Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems
aa r X i v : . [ m a t h . D S ] J a n Open Problems, Questions, and Challenges inFinite-Dimensional Integrable Systems.
Alexey Bolsinov ∗ , Vladimir S. Matveev † , Eva Miranda ‡ , Serge Tabachnikov § Abstract
The paper surveys open problems and questions related to different aspects of integrable sys-tems with finitely many degrees of freedom. Many of the open problems were suggested by theparticipants of the conference “Finite-dimensional Integrable Systems, FDIS 2017” held at CRM,Barcelona in July 2017.
Keywords: integrable system, Poisson manifold, Lagrangian fibration, bi-Hamiltonian system, Ni-jenhuis operator, integrable geodesic flows, integrable billiards, pentagram maps, quantisation.
Contents ∗ Department of Mathematical Sciences, Loughborough University, LE11 3TU, UK and Faculty of Mechanics andMathematics, Moscow State University, Moscow 119991, Russia
[email protected] † Institut f¨ur Mathematik, Fakult¨at f¨ur Mathematik und Informatik, Friedrich-Schiller-Universit¨at Jena, 07737 Jena,Germany [email protected] ‡ Department of Mathematics, Universitat Polit`ecnica de Catalunya and BGSMath, Barcelona, Spain, [email protected] § Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA [email protected]
Many different and formally nonequivalent definitions of finite dimensional integrable systems willbe used in this paper. The basic one is as follow: we consider a symplectic manifold ( N n , ω ), afunction H : N n → R on it, and the Hamiltonian system corresponding to it. It is integrable , if there2xist n − F , ..., F n − (called integrals) such that the n -functions H, F , ..., F n − commutewith respect to the Poisson structure corresponding to the symplectic structure, and their differentialsare linearly independent at almost every point of the manifold. Different generalization of this basicdefinition will be of course considered, for example, for Poisson manifolds (for a Poisson structure ofrank 2 n on a 2 n + r dimensional manifold we require the existence of r Casimirs F n , ..., F n + r − and n commuting integrals, see § N n ; then we require the existence of n functions whichcommute with respect to the Poisson structure corresponding to the symplectic structure and whichare preserved by the symplectomorphism.The theory of integrable systems appeared as a group of methods which can be applied to find exactsolutions of dynamical systems, and has grown in a separate field of investigation, which uses methodsfrom other branches of mathematics and has applications in different fields.The present survey collects open problems in the field of finite dimensional integrable systems, andalso in other fields, whose solution may influence the theory of integrable systems. We do hope thatsuch a list will be useful in particular for early stage researchers, since many problems in the list can, infact, be seen as possible topics of PhD theses, and also for people from other branches of mathematics,since for many problems in our list their solutions are expected to use ‘external’ methods.Many open problems in our list are fairly complicated and are possibly out of immediate reach; in thiscase we tried to also present easier special cases. But even in these cases we tried to keep the level ofproblems high – in our opinion, for most problems in our list, the solution deserves to be published.This list is motivated by the series of FDIS conferences organized every second year at differentlocations (in 2017, in Barcelona), and which is possibly the most important series of conferences inthe theory of finite dimensional integrable systems. Many problems in our list are suggested by theparticipants of this conference. Needless to say, the final selection of the material reflects the taste,knowledge, and interests of the authors of this article. Acknowledgements.
We are grateful to our colleagues who responded to our request to share openproblems and conjectures on finite-dimensional integrable systems with us: M. Bialy, D. Bouloc, H.Dullin, A. Glutsyuk, A. Izosimov, V. Kaloshin, R. Kenyon, B. Khesin, A. Pelayo, V. Retakh, V.Roubtsov, D. Treschev, S. V˜u Ngo. c, and N. Zung. We thank the referees of this paper for their usefulcomments and suggestions.A. B. is supported by the Russian Science Foundation (project no. 17-11-01303). V. M. is supportedby the DFG. E. M. is supported by the Catalan Institution for Research and Advanced Studies via anICREA Academia Prize 2016 and partially supported by the grants MTM2015-69135-P/FEDER and2017SGR932 (AGAUR). S.T. was supported by NSF grant DMS-1510055.3
Topology of integrable systems, Lagrangian fibrations, andtheir invariants
Recall that a finite dimensional integrable system on a symplectic manifold (
M, ω ) gives rise to a singular Lagrangian fibration whose fibres, by definition, are common level manifolds of the integrals F , . . . , F n . Our assumption that the integrals commute implies that nondegenerate fibers are La-grangian submanifolds, i.e., their dimension is half the dimension of the manifold, and the restrictionof ω to them vanishes. The word ‘singular’ means that we allow fibers which are not manifolds. Suchfibers typically contain points x ∈ M where the differentials dF ( x ) , . . . , dF n ( x ) are linearly dependent.Consider a singular Lagrangian fibration φ : M → B with compact fibers associated with a finite-dimensional integrable system. In general, the base B of this fibration is not a manifold, but can beunderstood as a stratified manifold whose strata correspond to different “singularity levels”. If weremove all singular fibers from M , then the remaining set M reg is foliated into regular Liouville tori,so that we have a locally trivial fibration M reg T −→ B reg over the regular part B reg of B . Notice that B reg is the union of strata of maximal dimension and, in general, may consist of several connectedcomponents.Recall that in a neighbourhood of each regular fiber (Liouville torus), one can define action-anglevariables. The actions can be treated as local coordinates on B . Since they are defined up to integeraffine transformations, they induce an integer affine structure on B reg . In a neighbourhood of a singularstratum, this affine structure, as a rule, is not defined but we still may study its asymptotic behaviourand think of it as a singular affine structure on B as a whole.The action variables are preserved under symplectomorphisms and, therefore, we may think of themas symplectic invariants of singular Lagrangian fibrations. How much information do they contain?Examples show that very often the action variables (equivalently, singular affine structure on B ) aresufficient to reconstruct completely the structure of the Lagrangian fibration up to symplectomor-phisms. The classical result illustrating this principle is Delzant theorem [60] stating that, in the caseof Lagrangian fibrations related to Hamiltonian torus actions, the base B is an affine polytope (withsome special properties) which determines the structure of the Lagrangian fibration (as well as thetorus action) up to a symplectomorphism. In particular, two Lagrangian fibrations M n T n −→ B and˜ M n T n −→ ˜ B are symplectically equivalent if and only if their bases B and ˜ B are equivalent as spaceswith integer affine structures.A similar statement still holds true for some Lagrangian fibrations with more complicated singularities(both non-degenerate and degenerate), see [65, 225, 39]. We believe that an analog of Delzant theoremholds true in a much more general situation. Ideally, such an analog could be formulated as thefollowing principle (it is not a theorem, as a counterexample is easy to construct!): Let φ : M → B and φ ′ : M ′ → B ′ be two singular Lagrangian fibrations. If B and B ′ are affinelyequivalent (as stratified manifolds with singular affine structures), then these Lagrangian fibrations arefiberwise symplectomorphic. uestion 2.1. Under which additional assumptions does this principle become a rigorous theorem?
In our opinion, this is an important general question, which is apparently quite difficult to answer infull generality. Some more specific questions could be of interest too.Let φ : M → B be an almost toric fibration (see [152, 204]), which means, in particular, that itssingularities are all non-degenerate and can be of elliptic and focus type only. Consider a typicalsituation when the base B of such a fibration is a two-dimensional domain with boundary (havingsome ‘corners’) and some isolated singular points of the focus type. This domain is endowed with aninteger affine structure, having singularities at focus points. Problem 2.2.
Consider two Lagrangian fibrations φ : M → B and φ ′ : M ′ → B ′ . Assume that B and B ′ are affinely equivalent in the sense that there exists an affine diffeomorphism ψ : B → B ′ . Isit true that under these assumptions the corresponding Lagrangian fibrations are symplectomorphic? In the local and semi-global setting, we may ask a similar question for a Lagrangian fibration in a neigh-bourhood of a singular point or of a singular fiber. According to the Eliasson theorem [72, 166, 169],in the non-degenerate case, there are no local symplectic invariants. In other words, non-degeneratesingular points of the same topological (algebraic) type are locally fiberwise symplectomorphic. Thecase of non-degenerate singular fibers has been discussed in [65, 225, 185, 186] for basic singularitiesof elliptic, hyperbolic, and focus types. One of the main observations made in these papers is thatevery action variable can be written in the form I = I sing + I reg , where the singular part I sing is thesame for all singularities of a given topological type (due to Eliasson’s result) and I reg is a smoothfunction which can be arbitrary. This regular part (or its Taylor expansion) in many cases may serveas a complete symplectic invariant in semi-global setting.Let us now discuss what happens in the case of degenerate singularities. Problem 2.3.
Do local symplectic invariants exist for diffeomorphic degenerate singularities? Howmany and of what kind are they? This question makes sense even in the simplest case of one-degree-of-freedom systems.
As we know from concrete examples of integrable systems, many degenerate singularities are stablein the sense that they cannot be destroyed by a small perturbation of the system [25, 85, 124]. Theproblem of description of stable degenerate singularities is very important on its own right, but formany degrees of freedom, it is quite complicated. However, in the case of two degrees of freedom(perhaps depending on a parameter), there is a number of well-known examples of stable bifurcations(e.g., integrable period-doubling [25] and Hamiltonian Hopf bifurcation [70]). It would be interestingto clarify the situation with asymptotic behaviour of angle variables at least for them.V. Kalashnikov [124] has described all stable singularities of rank 1 for Hamiltonian systems of twodegrees of freedom.
Problem 2.4.
Describe the symplectic invariants of stable singularities from [124]. For such singu-larities, one of the action variables, say I , is smooth, and the other I is singular. Is it true that I and I are sufficient for the symplectic classification? ere is an almost equivalent version of this question. Assume that we have two functions H and F that commute with respect to two different symplectic structures ω and ω and define (in both cases)a stable singularity from [124]. Assume that the action variables I and I are the same for ω and ω . This condition is equivalent to the relation I γ α = I γ α for any cycle γ on any regular fiber L f,h = { F = f, H = h } and appropriately chosen -forms α i such that dα i = ω i , i = 1 , . Is it true that, under these conditions, there exists a smooth map ψ thatpreserves the functions F and H and such that ψ ∗ ( ω ) = ω ? In the simplest case the answer is in[39]. Question 2.5.
Assume that we know explicit formulas for the action variables I , . . . , I n so thatwe are able to analyse their asymptotic behaviour in a neighbourhood of a singular fiber. Can werecover the topology of this singularity from the asymptotic behaviour (or at least to distinguish betweendifferent types of singularities)? For example, we know that, in the case of non-degenerate hyperbolicsingularities, the singular part I sing is of the form h ln h + . . . . Is this property characteristic fornon-degenerate hyperbolic singularities? The case of one degree of freedom is understood in [99]. For degenerate singularities, this question becomes, of course, more interesting and important.Another interesting problem would be a description of singularities, both in local and semi-globalsetting, which may occur in algebraically integrable systems. There is no commonly accepted definitionof algebraic integrability (see discussion on this subject and relevant definitions in [220]) and the answermay depend on the approach used. In the most natural setting, algebraic integrability assumes thatthe commuting functions F , . . . , F n are polynomials and one can naturally complexify all the relatedobjects (including the phase space, symplectic or Poisson structure, and time) so that the integrablesystem under consideration becomes the real part of a certain complex integrable system. The crucialadditional condition on this complex system is that its general complex fibers are isomorphic to affineparts of Abelian varieties (see [220]) which imposes rather strong topological conditions on singularities.For instance, in the case of one degree of freedom, it implies that the Milnor number of a singularpoint cannot be greater than 2. In other words, singularities of algebraically integrable systems areexpected to be very special. Problem 2.6.
Describe all the topological types of singularities that may appear in algebraically inte-grable systems with a small ( ≤ degree of freedom. Next, describe the symplectic invariants of suchsingularities. We expect that solving local and semi-global problems mentioned above will simplify the passage toglobal results. Some open problems related to classification of integrable systems in global setting arestated below.Recall that an integrable system F = ( J, H ) is called semitoric if all of its singularities are nondegen-erate and of either elliptic or focus types (hyperbolic singularties are forbidden) and, in addition, J isproper and generates a Hamiltonian S -action on the manifold.6 roblem 2.7 ( ´A. Pelayo) . Extend the classification of semitoric systems F = ( J, H ) in [185, 186] toallow for F having non-degenerate singularities with hyperbolic blocks (a version of this problem wasmentioned in various forms in [188]). The first step in this direction is in Dullin et al [70], see also a general approach to the symplecticclassification problem in Zung [238].Note that, in contrast with the semitoric case, even the topology of a small neighbourhood of a singularfiber can be quite complicated, see e.g. [23, 37, 121, 151, 158, 159, 160, 183, 236, 237, 238].There are also a number of interesting problems at the intersection of integrable systems and Hamil-tonian compact group actions, the following being one of them.
Problem 2.8 ( ´A. Pelayo) . Consider a compact connected n -dimensional symplectic manifold M ,endowed with a Hamiltonian ( S ) n − -action; these are called complexity one spaces . Consider anintegrable system f , . . . , f n on M where ( f , . . . , f n − ) : M → R n − is the momentum map of theHamiltonian ( S ) n − -action. Suppose that the singularities of the integrable system are non-degenerateand also they do not contain hyperbolic blocks. Study how the invariants of the complexity one spaceare related to the invariants of the semitoric system. For recent progress in the case 2 n = 4 see Hohloch et al [116].Recently there have been many works on integrable systems which can be obtained by the method oftoric degenerations. In particular, many well-known integrable systems can be constructed this way,including the bending flow of 3D polygons Foth et al [77] and the Gelfand-Cetlin system Nishinou et al [174]. See also [113, 130, 175]. A common feature of integrable systems coming from toric degenerationsis that their base space (the space of fibers of the momentum map) is a convex polytope, which impliesthat the singular fibers are quite special. Problem 2.9 (N. T. Zung) . Study the topology and geometry of these singular fibers and their smallneighbourhoods.
First results in this directions are due to D. Bouloc [40], where the bending flows of 3D polygons and thesystem on the 2-Grasmannian manifold were studied, and by Bouloc et al [41] for the Gelfand-Cetlinsystem.The natural conjecture is that, in most cases, the degenerate singular fibers are still smooth manifoldsor orbifolds and, moreover, they are isotropic (or even Lagrangian). They represent a new interestingclass of singularities in integrable systems for which almost all traditional questions remain open(symplectic invariants, normal forms, stability).Another problem is related to the so called SYZ model of mirror symmetry. In the case of Lagrangianfibrations on Calabi-Yau manifolds (see for example [47, 96, 97]) of complex dimension 3 (real dimension6) there are 1-dimensional families of singularities of focus-focus type (which are well-understood from Another common spellings are Zetlin and Zeitlin.
Problem 2.10 (N. T. Zung) . Give a clear description of these special singular fibers.
By a natural Hamiltonian system we understand a Hamiltonian system constructed as follows. Takea manifold M = M n equipped with a metric g = g ij and a function U : M → R , then consider theHamiltonian system on the cotangent bundle T ∗ M with the Hamiltonian given by H = K + U = g ij p i p j + U. (3.1)Here p = ( p , . . . , p n ) denote the momenta (i.e., the coordinates on T ∗ x M associated to local coordinates x = ( x , ..., x n ) on M ) and g ij is the inverse matrix to g ij . Summation over repeating indices isassumed, and the symplectic structure on T ∗ M is canonical.Of course, H itself is an integral for such a system, which we call trivial. Also all functions of H willbe considered as trivial integrals. In this section we discuss the existence of an integral F (or a family of integrals F , ..., F n − ) whichis polynomial of some fixed degree in momenta. The coefficients of this polynomial can, and usuallydo, depend on the position of the point x ∈ M , and in local coordinates they are functions on M . By[142], if the manifold M and the Hamiltonian H are smooth, these functions are automatically smoothtoo.This is one of the oldest part of the theory of integrable systems. Indeed, most, if not all, classicallyknown integrable systems admit integrals which are polynomial in momenta. For brevity, in whatfollows we will sometimes refer to them as polynomial integrals , meaning polynomial in momenta .A generic natural integrable system admits no nontrivial polynomial integral even locally, see [142]. Theexistence of such an integral is therefore a nontrivial differential-geometrical condition on the metric g and the function U , which actually appears naturally in some topics in differential geometry. Therelation to differential geometry becomes even clearer in the important special case, when the function U is identically zero. Such Hamiltonian systems are called geodesic flows . It is well known, and is easyto check, that polynomial integrals of geodesic flows are in natural one-to-one correspondence with theso-called Killing tensors.Note however that the condition that U is zero does not make the investigation of integrability easier.Indeed, by Maupertuis’s principle (see e.g. [21]), for each fixed value h of the energy H = K + U ,8he restriction of the Hamiltonian system to the isoenergy level { ( x, p ) ∈ T ∗ M | H ( x, p ) = h } is closely related (= is the same up to a re-parameterisation) to the geodesic flow of the metric g new = ( h − U ( x )) g . In the latter formula we assume that h = U ( x ), which can be achieved, forexample, if the energy value h is big enough. Polynomial in momenta integrals for H generate thenpolynomial in momenta integrals for the geodesic flow of g new . By Kolokoltsov [138], no metric on a closed surface of genus ≥ ≥ S Torus T Degree 1 All is known All is knownDegree 2 All is known All is knownDegree 3 Series of examples Partial negative resultsDegree 4 Series of examples Partial negative resultsDegree ≥ et al. [59] and Agapov et al. [1]. These papers study integrability, in the class ofpolynomial integrals, of natural Hamiltonian systems on T under the condition that the metric g isflat. It is proved that the existence of a nontrivial polynomial integral of degree m ≤ m = 5 under the additional assumption of real analyticity.As for “Degree ≥
5” line of the table, let us first mention the results of Kiyohara [137] who, for each d ≥
3, constructed a family of metrics on S whose geodesic flows admit a polynomial integral ofdegree d , but do not admit any nontrivial integrals of degree 1 or 2. Unfortunately, for the examplesconstructed in [137], it is not clear whether they might have nontrivial integral of “intermediate”degrees d ′ (i.e., 2 < d ′ < d ) or not. The integral F of degree d constructed in [137] is irreducible, butone may still imagine that there exists an additional polynomial integral of a smaller degree which isindependent of both H and F . The point is that the metrics constructed in [137] are Zoll metrics ,i.e., all of their geodesics are closed. This means that the geodesic flows are superintegrable (see thediscussion in § roblem 3.1. Complete the above table: (1) construct new examples of natural Hamiltonian systemson closed two-dimensional surfaces admitting polynomial integrals, describe and, if possible, classifythem; (2) prove, if possible, nonexistence of such integrals (perhaps under some additional assump-tions).
Of course, this general problem is very complicated, and a solution of certain special cases (some ofthem are listed below) is interesting. Even a construction of one new integrable example on a closedsurface deserves to be published!Let us list some conjectures/special cases/additional assumptions which were studied in relation toProblem 3.1.
Conjecture 3.2 ([21]) . If the geodesic flow of a Riemannian metric on the torus T admits a nontrivialintegral that is polynomial of degree 3 in momenta, then the metric admits a Killing vector field. An easier version of this conjecture, which is also interesting, is as follows:
Conjecture 3.3.
If a natural Hamiltonian system with a nonconstant potential on T admits anintegral that is polynomial of odd degree in momenta, then it admits a linear integral. If it admits anontrivial integral of even degree, then it admits a nontrivial quadratic integral. Resent results in this conjecture include [55]. Even the case when the integral has degrees 3 and 4 isnot solved! Yet an easier version is the following conjecture:
Conjecture 3.4 (Communicated by M. Bialy) . If a natural Hamiltonian system with a nonconstantpotential on a flat two-torus T ( i.e., g has zero curvature ) admits a polynomial integral of odd degree,then it admits a linear integral. If it admits a nontrivial integral of even degree, then it admits anontrivial quadratic integral. As explained above, the cases d = 3 , , U is a trigonometric polynomial, the problem was solved by N. Denisova and V. Kozlov in[58].Analogs of Conjectures 3.2, 3.3, 3.4 are also interesting if the metric g is semi-Riemannian (i.e., anindefinite metric of signature (1 , T whose geodesic flows admit integrals which are linearor quadratic in momenta were described in [163] and, combining this result with [28], one obtains adescription of all natural Hamiltonian systems on semi-Riemannian 2-tori with linear and quadraticintegrals.As seen from the discussion above, in the case of the sphere S , almost nothing is known about integralsof degree d ≥
5, which suggests the following open problem:
Problem 3.5.
Construct a natural Hamiltonian system with a nonconstant potential on S that admitsa nontrivial polynomial integral of degree and does not admit any nontrivial integrals of smallerdegrees. T ∗ M , namely such that H := K + L + U, where K and U are as in (3.1), and L is a linear in momenta function (sometimes referred to asmagnetic terms); the solutions of this Hamiltonian system describe trajectories of charged particlesin a magnetic field. In this case, only few explicit examples are known (e.g., those coming fromKovalevskaya, Clebsh, and Steklov integrable cases in rigid body dynamics, and also from Veselovasystem and Chaplygin ball from non-holonomic integrable systems), so any new example would be aninteresting result. Notice that even an explicit local description of such systems in the simplest caseof quadratic integrals in dimension two remains an open problem (we refer to [157] by V. Marikhin etal and [193] by G. Pucacco et al for the latest progress in the area).Note that in this case one can consider integrability on the whole cotangent bundle, as it was done,for example, in [157], or on one energy level as, for example, in [32, § et al [45], see § §§ In most known examples, the integrals of natural systems are either related to Hamiltonian reductionleading to integrable systems on Lie algebras (which may possess nontrivial polynomial integrals ofarbitrary degree, see e.g. [26]), or the integrals are quadratic in momenta and the quadratic terms aresimultaneously diagonalisable. There are also examples with polynomial integrals of higher degreescoming from physical models, such as the Toda lattice and Calogero–Moser systems. Generally, onesimply needs more examples, which either are interesting to mathematical physics or “live” on closedmanifolds.
Problem 3.6.
Construct new examples of natural Hamiltonian systems on higher dimensional mani-folds which are integrable in the class of integrals polynomial in momenta.
Another interesting direction of research might be relevant to General Relativity. Recall that the mostfamous solutions of the Einstein equation, Schwarzschild and Kerr metrics, have integrable geodesicflow.
Question 3.7.
Construct stationary axially symmetric 4-dimensional Einstein metrics admittingKilling tensors of higher order.
The condition “stationary axially symmetric” simply means the existence of two commuting vectorfields, one of which is space-like and the other time-like.11et us also touch upon two problems related to the homogeneous case. Recall that a scalar invariantof order k of a Riemannian manifold ( M, g ) is a function on M , constructed from covariant derivativesof order k or less of the curvature tensor R or, equivalently, an algebraic expression in the componentsof covariant derivatives of R (of order ≤ k ) which transforms as a function under coordinate changes.We say that ( M, g ) is k - homogeneous , if all its scalar invariants up to order k are constants.It is known that, for a sufficiently big k , any k -homogeneous manifold is automatically locally homoge-neous, i.e., admits a locally transitive isometry group. As it follows from Singer [201], k = n ( n +1) / n is the dimension of the manifold, is sufficiently big in this sense; later better estimations werefound in [192, 54]. Question 3.8 (Gilkey [86]) . In the Riemannian case, is every -homogeneous manifold locally homo-geneous? The question was positively answered for dimensions 3 ,
4. The natural analogs of this conjecture formetrics of non-Riemannian signature are wrong. Notice that, in terms of polynomial integrals, localhomogeneity is equivalent to the existence of linear integrals of the geodesic flow that at each pointspan the tangent space.
Question 3.9.
In a symmetric space, is every Killing tensor a sum of symmetric products of Killingvectors? Equivalently, is it true that the algebra of all polynomial integrals of the geodesic flow in asymmetric space is generated by linear integrals?
Note that for general metrics the answer is clearly negative: there exist metrics admitting nontrivialKilling tensors and no Killing vectors. For constant curvature spaces, the answer is positive [212].
Recall that a natural Hamiltonian system is superintegrable , if it admits more functionally independentintegrals (of some special form) than the dimension of the manifold (in addition, one usually assumesthat common level surfaces of these integrals are isotropic submanifolds in T ∗ M ). A system is called maximally superintegrable , if the number of functionally independent integrals is 2 n −
1, which is themaximal number for an n -dimensional manifold.An interesting and well-studied class of superintegrable natural Hamiltonian systems consists of geodesicflows of Zoll metrics. Recall that a Riemannian metric (on a closed manifold) is called a Zoll metric ifall its geodesics are closed, see e.g. [11]. In this case, the Hamiltonian flow can be viewed as a symplec-tic action of S , the orbit space is an orbifold, and any function defined on this orbifold is an integral.Notice that locally, near almost every point, each natural Hamiltonian system is superintegrable. It istherefore standard to consider superintegrability in the class of integrals of a special form, typically,polynomial in momenta. Then, superintegrability is a nontrivial condition on the system, even locally.In dimension two, superintegrable systems with quadratic integrals were locally essentially describedin [57]. Superintegrable systems with one linear and one cubic integral were described in [162]; the12ethods can be generalised for the case when one integral is linear and the second of arbitrary degree,see e.g. [218]. Problem 3.10.
Construct a natural Hamiltonian system on the 2-sphere with a nonconstant potentialwhich is superintegrable by integrals of degree ≥ and admits no nontrivial integral of degree one andtwo. In higher dimensions, partial answers were obtained for conformally flat metrics only and are wellelaborated only in dimension 3 for quadratic integrals, see e.g. [125, 126]. The ultimate goal is,of course, to describe all superintegrable (in the class of polynomial in momenta integrals) naturalHamiltonian systems. As the first nontrivial problem we suggest the following question:
Question 3.11.
Does there exist a non-conformally flat metric on the sphere S n , n > , whose geodesicflow is maximally superintegrable (in the class of integrals which are polynomial in momenta)? Iterations of a smooth map f : M → M yield a discrete-time dynamical system on a manifold M .More precisely, the evolution of a point x ∈ M is given by repeated applications of the map f : x f ( x ) f ◦ ( x ) . . . f ◦ k ( x ) . . . Thus the time takes non-negative integral values. If the map is invertible, then the orbit of a point isalso defined for negative values of k .By Liouville integrability of such a system one means the following: • The manifold M n carries an f -invariant symplectic structure ω ; • There exist smooth functions (integrals) I , . . . , I n on M , functionally independent almost every-where, invariant under the map f , and Poisson commuting with respect to the Poisson structureinduced by ω .See [221] for definitions and examples.A modification of this definition is given by a Poisson manifold M p +2 q whose Poisson structure hascorank p . Then Liouville integrability of a map f : M → M means that the Poisson structure is f -invariant, and f possesses almost everywhere functionally independent integrals I , . . . , I p + q , of whichfirst p are Casimir functions (the functions that Poisson commute with everything).A recent popular example is the pentagram map introduced in 1992 by R. Schwartz [200], a projectivelyinvariant iteration on the moduli space of projective equivalence classes of polygons in the projective13lane. This map is Liouville integrable [178, 179]; its algebraic-geometric integrability was establishedin [203]. The integrability of the pentagram map is closely related with the theory of cluster algebras,see, e.g., [74, 79, 87, 88, 94]. The billiard system describes the motion of a free particle in a domain with elastic reflection off theboundary. Let Ω ⊂ R be a strictly convex domain bounded by a smooth curve γ . A point moves insideΩ with unit speed until it reached the boundary where it reflects according to the law of geometricaloptics: the angle of reflection equals the angle of incidence, see, e.g., [206].An extension to higher dimensions and, more generally, to Riemannian manifolds with boundary isimmediate: at the moment of reflection, the normal component of the velocity changes sign, whereasthe tangential component remains the same.The phase space of the billiard system consists of oriented lines (or oriented geodesics), endowed withits canonical symplectic structure, obtained by symplectic reduction from the symplectic structure ofthe cotangent bundle of the ambient space. These oriented lines are thought of as rays of light; thebilliard map takes the incoming ray to the reflected one.In dimension two, an invariant curve of the billiard map can be thought as a 1-parameter family oforiented lines. Their envelope is called a billiard caustic : a ray tangent to a caustic remains tangentto it after the reflection.A variation is the billiard system in the presence of a magnetic field. In the plane, or on a Riemanniansurface, a magnetic field is given by a function B ( x ), and the motion of a charged particle is describedby the differential equation ¨ x = B ( x ) J ˙ x , where J is the rotation of the tangent plane by π/
2. Thisequation implies that the speed of the particle remains constant (the Lorentz force is perpendicular tothe direction of motion). In the Euclidean plane, if the magnetic field is constant, the trajectories arecircles of a fixed radius, called the Larmor circles.In general, a magnetic field on a Riemannian manifold M is a closed differential 2-form β , and themagnetic flow is the Hamiltonian flow of the usual Hamiltonian | p | / T ∗ M with respect to the twisted symplectic structure dp ∧ dq + π ∗ ( β ), where π : T ∗ M → M is the projectionof the cotangent bundle on the base.Another generalization are the Finsler billiards: the reflection law is determined by a variationalprinciple saying that the trajectories extremize the Finsler length (which is not necessarily symmetric),see [108]. Magnetic billiards are particular cases of Finsler billiards, see [208] or [19].Still another variation is the outer billiard, an area preserving transformation of the exterior of a closedconvex plane curve. Like inner billiards, outer ones can be defined in higher dimensions: a curve inthe plane is replaced by a closed smooth strictly convex hypersurface in linear symplectic space, andthe outer billiard transformation becomes a symplectic mapping of its exterior.14uter billiards can be defined in two-dimensional spherical and hyperbolic geometries as well. In thespherical case, the inner and outer billiards are conjugated by the spherical duality. See [63] for asurvey.Let us also mention two other non-conventional billiard systems: projective billiards [207] and sym-plectic billiards [2]. It is a classical fact (see for instance [206] and references therein) that the billiard in an ellipse isintegrable: the interior of the table is foliated by caustics, the confocal ellipses. Likewise, the billiardinside an ellipsoid is Liouville integrable, see, e.g., [64, 209].The following is known as the Birkhoff conjecture, see [191].
Conjecture 4.1.
If a neighbourhood of the boundary of the billiard table Ω ⊂ R is foliated by caustics,then the billiard curve γ is an ellipse. We also formulate a generalized conjecture: if an open subset of the convex billiard table is foliated bycaustics, then this table is elliptic .A polynomial version of Birkhoff’s conjecture concerns the case when the integral of the billiard map ispolynomial in the velocity. In the case of outer billiards, one modifies these conjectures in an obviousway (since outer billiards are affine-equivariant, there is no difference between ellipses and circles, andcircles are integrable due to symmetry).The state of the art of the (smooth) Birkhoff conjecture is as follows. The special case of the conjecturein which the caustics of the billiard trajectories foliate Ω, except for one point, was proved by M. Bialy[13]; later he extended this result to the elliptic and hyperbolic planes [16]. Bialy also proved a versionof this result for magnetic billiards in constant magnetic field on surfaces of constant curvature [14].Recently, substantial progress toward Birkhoff’s conjecture was made by V. Kaloshin and his co-authors[5, 127, 117]: they proved the Birkhoff conjecture for small perturbations of ellipses. The general caseremains open, and it is one of the foremost problems in this field.
Problem 4.2.
Assume that the exterior of an outer billiard table is foliated by invariant curves. Provethat the table is an ellipse.
The integral-geometric methods used by Bialy in his proofs fail due to non-compactness of the phasespace.The following is a generalization of Birkhoff’s conjecture.
Conjecture 4.3. (S. Tabachnikov)
Let γ ⊂ RP be a smooth closed strictly convex curve, and let F be a smooth foliation by convex closed curves in its exterior neighbourhood, including γ as a leaf. The ntersection of every tangent line to γ with the leaves of red F define a local involution on this line,the point of tangency being a fixed point. Suppose that these involutions are projective transformationsfor all tangent lines of γ . Then γ is an ellipse, and the foliation consists of ellipses that form a pencil. The converse statement, that the intersections of a line with the conics from a pencil define a projectiveinvolution on this line, is a classical Desargues theorem.A particular case of the above conjecture, when the involutions are isometries, yields the Birkhoffconjecture for outer billiards. In the generalized form, this conjecture implies, via projective duality,a version of Birkhoff conjecture for projective billiards and hence for the usual inner billiards.The above formulated conjectures and problems have algebraic versions, and much progress has beenmade in this area recently. For inner billiards, the polynomial Birkhoff conjecture was proved in[18, 90], including the cases of non-zero constant curvature, and for outer billiards in the affine plane– in [91].
Problem 4.4.
Prove the algebraic version of Birkhoff conjecture for outer billiards in a non-Euclideansurface of constant curvature.
The next problem is a generalization of the previous one.
Problem 4.5.
Prove Conjecture 4.3 it the case that the foliation admits (i) a rational, (ii) an algebraicfirst integral.
Algebraic non-integrability of magnetic billiards was studied by Bialy and Mironov in [17]: they provedthat a non-circular magnetic Birkhoff billiard is not algebraically integrable for all but finitely manyvalues of the magnitude of the (constant) magnetic field. It is plausible that the result holds for everyvalue of the strength of the magnetic field, but this remains an open problem.Consider a convex billiard table having the symmetry of an ellipse. Such a billiard has a 2-periodicorbit along the y -axis. Problem 4.6.
Is it possible to choose the domain so that the dynamics of the corresponding billiardmap are locally (near the 2-periodic orbit) conjugated to the dynamics of the rigid rotation throughsome angle α ? V. Schastnyy and D. Treschev [199] provide numerical evidence that this is possible if α/π is a Dio-phantine number. A p/q -rational caustic corresponds to an invariant circle of the billiard map that consists of periodicpoints with the rotation number p/q . For example, a curve of constant width has a 1 / p/q -rational caustics for all sufficiently small values of p/q – this follows from the Arnold-Liouvilletheorem. An ellipse possesses p/q -rational caustics for all p/q < / p/q -rational caustic for a fixed value of p/q ;see [84] for outer billiards. Question 4.7.
Are there plane billiards, other than ellipses, that possess rational caustics with twodifferent values of the rotation numbers? Same question for outer billiards.
Kaloshin and J. Zhang [129] studied the case of 1 / / q ≥ q -rationally integrable is it possesses p/q -rational caustics for all 0 < p/q < /q (see [117]). Conjecture 4.8. (V. Kaloshin) A q -rationally integrable billiard is elliptic. The next conjecture is motivated by a theorem of N. Innami’s: if there is a sequence of convex causticswith rotation numbers tending to 1 /
2, then the billiard table is an ellipse [120] (see [3] for a simplerproof). An invariant circle of the billiard map is called rotational if it consists of periodic points.
Conjecture 4.9. (M. Bialy)
Assume that a plane billiard has a family of rotational invariant circleswith all the rotation numbers in the interval (0 , / . Then the billiard table is an ellipse. A variation on Innami’s result is the following question, suggested by V. Kaloshin:
Question 4.10.
Suppose that a billiard table has a sequence of convex caustics with rotation numbersconverging to some number ω ∈ (0 , / . Does it imply that the table is an ellipse? The previously discussed rotation numbers, 0 and 1 /
2, are clearly special.One may generalize the notion of caustic for higher-dimensional billiards as a hypersurface such thatif a segment of the billiard trajectory is tangent to it, then the reflected segment is again tangent tothis hypersurface. However, this notion is very restrictive: if the dimension of the Euclidean space isat least three, the only convex billiard table that possess caustics are ellipsoids [10, 98].
Question 4.11. (M. Bialy)
Are there multi-dimensional convex billiards, other than ellipsoids, havinginvariant hypersurfaces in the phase space? Same question for multi-dimensional outer billiards.
A common feature of families of integrable maps is that they commute (Bianchi permutability).17onsider two confocal ellipses: the billiard reflections define two transformations on the set of orientedlines that intersect both ellipses. These transformations commute, see [209]. The converse was recentlyproved by A. Glutsyuk [89], as a consequence of his classification of 4-reflective complex planar billiards.For outer billiards, an analogous theorem is that if the outer billiard transformations on two nestedcurves commute then the curves are concentric homothetic ellipses [205]. It would be interesting tofind a proof of Glutsyuk’s result via a more geometric approach of [205].In higher dimensions (greater than two), the billiard transformations, induced by two confocal ellip-soids, commute as well (see, e.g., [64]).
Problem 4.12. (A. Glutsyuk, S. Tabachnikov [210])
Given two nested closed convex hypersurfaces,assume that the billiard transformations on the set of oriented lines that intersect both hypersurfacescommute. Prove that the hypersurfaces are confocal ellipsoids.
The above problem also has analogs in the elliptic and hyperbolic geometries. Namely, one can definequadrics in the elliptic and hyperbolic geometries: a quadratic hypersurface of the unit sphere is, bydefinition, its intersection with a quadratic cone, and a similar definition applies to the hyperboloidmodel of the hyperbolic geometry. It is known that the billiard maps in quadrics are completelyintegrable in the geometries of constant curvature, see, e.g., [206].
We already mentioned the pentagram map. This map sends a (generic) polygon in the projectiveplane to a new polygon whose vertices are the intersection points of the consecutive short diagonals ofthe original polygon. The map extends to twisted n -gons, bi-infinite collections of point ( v i ) satisfying v i + n = M ( v i ) for all i , where M is a projective transformation. The pentagram map also commuteswith projective transformations (since it only involves points and lines, and the operations of connectingpairs of points by lines and intersecting lines), hence the pentagram map is defined on the moduli spaceof projective equivalence classes of polygons.Numerous generalizations of the pentagram map were introduced and studied, in particular, in [131,132, 133, 156]. Following [132], we describe one class of such generalized pentagram maps and presentsome problems and conjectures.Let ( v i ) be a twisted polygon in RP d . Fix two ( d − I = ( i , . . . , i d − ) , J = ( j , . . . , j d − ) . Define the hyperplanes P k = Span( v k , v k + i , v k + i + i , . . . , v k + i + ... + i d − ) , and consider the new point T I,J ( v k ) = P k ∩ P k + j ∩ P k + j + j ∩ . . . ∩ P k + j + ... + j d − . T I,J . In this notation, the usual pentagram mapis T (2) , (1) .For some of these maps, for example, T (2 , , (1 , , T (2 , , (1 , , and T (3 , , (1 , , complete integrability hasbeen established [131, 133, 156]. Other cases were studied numerically in [132].This study was based on the height criterion [110, 95]. The height of a rational number p/q , givenin the lowest terms, is max( | p | , | q | ). In appropriate coordinates, T I,J is a rational map defined over Q . One considers the growth of the maximal heights of the coordinates under iterations of the map.For integrable maps, one expects a slower growth, and for non-integrable ones – a faster one. Morespecifically, a linear growth of the double logarithm of this height is a numerical indication of non-integrability.In particular, these experiments suggested that the maps T (2 , , (1 , , T (1 , , (1 , , T (1 , , (1 , , T (2 , , (2 , , are integrable, whereas the maps T (1 , , (3 , , T (1 , , (1 , , T (2 , , (1 , , T (2 , , (1 , , T (3 , , (1 , are not integrable. Conjecture 4.13. (B. Khesin, F. Soloviev)
The maps T I,I are integrable for all multi-indices I . Khesin and Soloviev also conjectured that all the maps T I,J (and even a more general type of maps,called universal pentagram maps) are Hamiltonian, that is, preserve a certain non-trivial Poissonstructure. See the recent paper by A. Izosimov on this subject [123]. Some of the maps considered byKhesin and Soloviev are included in the family of transformations described in [88]; they are examplesof integrable cluster dynamics.
This section concerns dynamical systems, with continuous or discrete time, on associative but notcommutative algebras, so that the variables involved do not commute; the reader with an aversion tofree non-commuting variables may think of matrices of an arbitrary size: see, e.g., [177, 165]. In thiscase, one speaks about noncommutative integrability , although this term has another, quite different,meaning as well (when the integrals form a non-Abelian Lie algebra with respect to the Poissonbracket).The overarching problem is to develop a comprehensive theory that would include noncommutativecounterpart to the definition of integrability as given in Section 4.1. As of now, the geometric aspectsof the noncommutative theory are poorly understood, and one mostly deals with the formal, algebraicpart of the story. In what follows, we work with noncommutative Laurent polynomials or power seriesin several variables. 19ere is an example of a 2-dimensional noncommutative integrable map that illustrates the specificsand peculiarities of the noncommutative situation. This example was discovered by S. Duzhin and M.Kontsevich [139] and studied in [4, 232]: T : ( x, y ) ( xyx − , (1 + y − ) x − ) . Here x and y are free non-commuting variables and the map can be understood as a substitution inthe ring of Laurent polynomials C h x, y, x − , y − i .The map T preserves the commutator xyx − y − . In the commutative case, when xy = yx , the mapbecomes T : ( x, y ) ( y, (1 + y − ) x − ) , but the integral xyx − y − degenerates to a constant. On the other hand, in the commutative case,one has an integral I = x + y + x − + y − + x − y − , whose level curves are elliptic curves, and the map has an invariant area form ( xy ) − dx ∧ dy , so it iscompletely integrable in the sense of Section 4.1.In the noncommutative case, the Laurent polynomial I is transformed by T as follows I xIx − . Ifthe variables represent matrices (of an arbitrary size), then tr I k is an integral of the transformation T for every positive integer k . In the general case, when x, y are formal variables, one defines trace onthe algebra of Laurent polynomials C h x, y, x − , y − i as an equivalence class of the cyclic permutationsof factors of a polynomial.On the other hand, in the intermediate (“quantum”) case, when xy = qyx , with q commuting witheverything, the map T has the integral I q = x + qy + qx − + y − + x − y − , whose q → I .The map T is a discrete symmetry of the system of differential equations˙ x = xy − xy − − y − , ˙ y = − yx + yx − + x − . Once again, we consider these differential equations in the formal sense, as Laurent series whosecoefficients depend on a parameter, and we are not concerned with the questions of existence anduniqueness of solutions.This system of ODEs admits a Lax presentation with a spectral parameter (that commutes witheverything); the entries of the Lax matrix are noncommutative Laurent polynomials in x and y , andthe conjugacy class of the Lax matrix does not change under the map T .The map T belongs to a family of maps T k : ( x, y ) ( xyx − , (1 + y k ) x − ) . k ≥
1, these maps satisfy the
Laurent phenomenon : both components of all the iterations belongto the noncommutative ring of Laurent polynomials in x, y (for k ≥
3, the maps T k are not integrable).This theorem, also conjectured by S. Duzhin and M. Kontsevich [139], was first proved in [217] and,independently, in [61], and later, by different methods, in [9, 150] (these papers also tackle differentvariations of the S. Duzhin and M. Kontsevich conjectures). In the commutative setting, the Laurentphenomenon is well known in the theory of cluster algebras and is related with integrability of clusterdynamical systems.We now switch gears and discuss an open problem concerning noncommutative versions of the penta-gram maps described in the previous section. These maps belong to the realm of projective geometry,so one needs a noncommutative version of projective geometry and, in particular, of the cross-ratio ofa quadruple of points on the projective line.A fundamental notion of noncommutative algebra is quasi-determinant [80, 81, 82], a noncommutativeanalog of the determinant (quasi-determinants play a role in the study of noncommutative integrability,see [73, 62]). Using quasi-determinants, V. Retakh [194] defined the cross-ratio and described itsproperties. Let us present the relevant definitions.Let A be a 2 × n matrix whose entries a i and a i , i = 1 , . . . , n , are elements of an associative ring R over a field. For 1 ≤ i, j, k ≤ n , define the quasi-Pl¨ucker coordinates of the matrix Aq kij ( A ) = ( a i − a k a − k a i ) − ( a j − a k a − k a j ) , where we assume that all the inversions are defined (quasi-Pl¨ucker coordinates can be defined for k × n matrices as well).Let x, y, z, t be four 2-vectors with components in R . These vectors form the columns of a 2 × A , and one defines the cross-ratio [ x, y, z, t ] = q yzt ( A ) q xtz ( A ) . In the commutative case, assuming that the second components of the vectors equal 1, one obtains afamiliar definition of the cross-ratio (or, better said, one of its six definitions):[ x, y, z, t ] = ( z − x )( t − y )( z − y )( t − x ) . The non-commutative cross-ratio shares some properties with its commutative counterpart. For ex-ample, [
Gxλ , Gyλ , Gzλ , Gtλ ] = λ − [ x, y, z, t ] λ , where G is a 2 × R , and λ i ∈ R, i = 1 , , , leapfrog map .Let S − = ( S − , S − , . . . ) and S − = ( S , S , . . . ) be a pair of n -gons in the projective line. The leapfrogmap ( S − , S ) ( S, S + ) is defined as follows. For every i , there exists a unique projective involution that21ends the triple ( S i − , S i , S i +1 ) to the triple ( S i +1 , S i , S i − ); this transformation sends S − i to S + i (thatis, the point S − i “jumps” over S i , in the projective metric on the segment [ S i − , S i +1 ], and becomes S + i – thus the name of the map). In other words, one has [ S i − , S i , S i +1 , S − i ] = [ S i +1 , S i , S i − , S + i ].The theory of noncommutative cross-ratios implies that there exists a projective transformation (cid:0) S i − , S i , S i +1 , S − i (cid:1) (cid:0) S i +1 , S i , S i − , S + i (cid:1) if and only if( S i +1 − S i ) − (cid:0) S − i − S i (cid:1) (cid:0) S − i − S i − (cid:1) − ( S i +1 − S i − ) = r − ( S i − − S i ) − (cid:0) S + i − S i (cid:1) (cid:0) S + i − S i +1 (cid:1) − ( S i − − S i +1 ) r where r ∈ R is invertible. In the commutative case, this becomes the formula defining the leapfrogmap.It was proved in [79] that the leapfrog map is completely integrable: it has an invariant Poissonstructure and a complete collection of Poisson commuting integrals; in the appendix to [79], A. Izosimovconstructed a tri-Hamiltonian structure for this map. Problem 4.14. (V. Retakh) Establish complete integrability of the noncommutative version of theleapfrog map. Define noncommutative versions of the pentagram map and its higher-dimensionalanalogs and establish their complete integrability.Let us add that a version of the pentagram map was defined and studied for polygons in Grassmannians[75]. It is a further challenge to define a noncommutative version of these maps and to investigatetheir integrability.
Let M be a smooth manifold. A Poisson bracket on M is a bilinear (over R ) skew-symmetric operation { , } on the space of smooth functions C ∞ ( M ) which satisfies the Leibniz rule { f g, h } = f { g, h } + { f, h } g and the Jacobi identity { f, { g, h }} + { h, { f, g }} + { g, { h, f }} = 0for any f, g, h ∈ C ∞ ( M ). This operation turns C ∞ ( M ) into an infinite-dimensional Lie algebra.In local coordinates, a Poisson bracket can be written as { f, g } ( x ) = n X i.j A ij ( x ) ∂f∂x i ∂g∂x j , where A = (cid:0) A ij ( x ) (cid:1) is the corresponding Poisson tensor. If the rank of A is smaller than the dimensionof M , the basic definition of a finite-dimensional integrable systems given in Introduction needs a22mall modification. Namely, we will assume that the number of independent commuting first integrals(including the Hamiltonian H ) equals 12 (dim M + corank A )where the corank of A is taken at a generic point. This condition is equivalent to the fact that thesubspace spanned by the differentials of the first integrals in the cotangent space T ∗ x M at a genericpoint x ∈ M is maximal isotropic with respect to the Poisson tensor A (which is exactly the samecondition as what we use in the symplectic case).One of the most effective methods for constructing and studying integrable systems is based on thenotion of compatible Poisson structures, see for example [153, 154]. Recall that two Poisson brackets { , } and { , } on M are called compatible if any linear combination of them is again a Poissonbracket. The set of all non-trivial linear combinations of two compatible Poisson brackets { , } and { , } is called a Poisson pencil . At a fixed point x ∈ M , this pencil can be understood as a pencil of skew-symmetric bilinear forms J = { A λ = A + λB } λ ∈ ¯ C on the tangent space T ∗ x M (we may consider these forms up to proportionalityand formally set A ∞ = B ). To each pencil J , we can naturally assign its algebraic type. To explainwhat is meant by the algebraic type of J , we recall the Jordan-Kronecker decomposition theorem (see,e.g., [38, 211]) that provides a simultaneous canonical form for a pair of skew-symmetric forms.Consider a complex vector space V with a pair of bilinear forms A, B : V × V → C . For simplicity wewill assume that B is regular in the pencil J = { A λ = A + λB } , i.e., the rank of B is maximal withinthis pencil. Theorem 5.1.
By choosing an appropriate basis, A and B can be simultaneously reduced to a block-diagonal form: A A . . . A s , B B . . . B s (5.1) where the corresponding pairs of blocks have one of the following two possible types :1) Jordan type µ i -block : A i = (cid:18) − Id 0 (cid:19) , B i = (cid:18) J ( µ i ) − J ⊤ ( µ i ) 0 (cid:19) , where J ( µ i ) is a Jordan block of size k i × k i with eigenvalue µ i and Id is the identity matrix of thesame size ; 2) Kronecker block : A i = (cid:18) D − D ⊤ (cid:19) , B i = (cid:18) D ′ − D ′⊤ (cid:19) , here D and D ′ are k i × ( k i + 1) –matrices of the form: D = . . . . . . , D ′ = . . . . . . By the algebraic type of the pencil J = { A λ = A + λB } we mean the type of the above decompositionthat includes the number and sizes of Kronecker blocks and Jordan blocks, separately for each charac-teristic number µ i (ignoring, however, the specific values of µ i ’s). It follows from this definition that,in each dimension, there are only finitely many different algebraic types.Clearly, the algebraic type of a Poisson pencil may depend on a point x ∈ M but, in the real analyticcase, there always exists an open everywhere dense subset U ⊂ M of generic points with the samealgebraic type. We will refer to it as the algebraic type of the Poisson pencil on M .It is well known that many properties of a bi-Hamiltonian system essentially depend on the alge-braic type of the underlying Poisson pencil. For instance, if this pencil is Kronecker (i.e., there areonly Kronecker blocks in the Jordan-Kronecker decomposition), then the integrals of any related bi-Hamiltonian system are Casimir functions of the brackets forming the pencils, whereas in the caseof Jordan (symplectic) pencils the integrals are the eigenvalues of the so-called recursion operator(equivalently, characteristic numbers of the Jordan-Kronecker decomposition). For Poisson pencils ofa “mixed” algebraic type, the situation becomes more complicated. In particular, there might be nocanonical choice of integrals for associated bi-Hamiltonian systems. One of the most natural problemsin this context would be to develop a theory of bi-Poisson vector spaces, i.e., spaces endowed with apencil of skew symmetric forms. Such a theory would serve as a natural analog of symplectic linearalgebra in view of the role the latter plays in symplectic geometry and the theory of Hamiltoniansystems. Some of results in this direction can be found in [38, 235], but still there is a number offurther questions to be clarified. Below we list some of them.The automorphism group of a pencil J = { A λ = A + λB } is an algebraic linear group defined byAut( V, J ) = { φ ∈ End( V ) | A λ ( φ ( ξ ) , φ ( η )) = A λ ( ξ, η ) for all A λ ∈ J } . (5.2)Obviously, the structure of this group essentially depends on the algebraic type of J (see [235]). Problem 5.2.
Consider the action of
Aut(
V, J ) on V . Describe the partition of V into Aut(
V, J ) -orbits. More generally, describe the action of Aut(
V, J ) on the set of all k -dimensional subspaces U ⊂ V (orbits, invariants, fixed points). Notice that fixed points of this action (i.e., Aut(
V, J ) -invariant subspaces) are important, as they correspond to well-defined (co)distributions in the contextof Poisson pencils. A subspace L ⊂ V is said to be bi-Lagrangian if it is simultaneously isotropic with respect to both A and B and has maximal possible dimension (which is (dim V + corank J )). The set LG( V, J )of all bi-Lagrangian subspaces is called the bi-Lagrangian Grassmannian . A detailed description of24G(
V, J ) is an important open problem, see the discussion in [195]. It is important that bi-Lagrangiansubspaces always exist, so that LG(
V, J ) is never empty. In general, this is a projective algebraicvariety with rather non-trivial properties, which essentially depend on the algebraic type of the pencil J . In particular, there might exists bi-Lagrangian subspaces of different algebraic types. In otherwords, LG( V, J ) may consist of several orbits of the automorphism group Aut(
V, J ). Problem 5.3.
Find necessary and sufficient conditions for the bi-Lagrangian Grassmannian
LG(
V, J ) to be a smooth algebraic variety. Describe the partition of LG(
V, J ) into Aut(
V, J ) -orbits. A “non-linear analog” of the existence of bi-Lagrangian subspaces would be the existence of completesubalgebras
F ⊂ C ∞ ( M ) consisting of functions in involution with respect to all the brackets froma given Poisson pencil. By completeness in this case we mean that the subspace generated by thedifferentials of functions f ∈ F in the cotangent space T ∗ x M is maximal isotropic. It is natural to referto such a subalgebra (or to a collection of its generators) as a bi-integrable system.In the case of Kronecker pencils, such a subalgebra F is generated by the (local) Casimir functionsof the brackets in the pencil. It is well known that F so obtained is complete and commutative withrespect to all the brackets in the pencil ([20, 83]). Similar results in the symplectic case are dueto P. Olver [176] and H. Turiel [216]. In the mixed case (when the Jordan-Kronecker decompositionof a pencil includes both Kronecker and Jordan blocks), bi-integrable systems do exist locally in aneighbourhood of a generic point. However, to the best of our knowledge, this result has never beenpublished.In view of the above discussion on the partition of LG( V, J ) into different orbits, it would be naturalto distinguish bi-integrable systems of different algebraic types. More precisely, let f , . . . , f k be acomplete set of functions in bi-involution. We restrict our considerations to a small neighbourhood U of a generic point x ∈ M such that for each y ∈ U the Jordan-Kronecker type of the pencil defined on T ∗ y M by a pair of compatible Poisson brackets is the same. If we think of T ∗ y M as a bi-Poisson vectorspace, then the subspace L y = span { df ( y ) , . . . , df k ( y ) } ⊂ T ∗ y M is bi-Lagrangian and therefore can becharacterised by its algebraic type (i.e., by the type of the orbits of the automorphism group actionin the bi-Lagrangian Grassmannian to which L belongs). If this type remains the same for all points y ∈ U , we refer to it as the algebraic type of the bi-integrable system f , . . . , f k . Problem 5.4.
Do bi-integrable systems exist for each algebraic type?
This question is local in the sense that bi-integrable systems in question “live” in a small neighbourhoodof a generic point x ∈ M where the algebraic type of a given Poisson pencil is locally constant. However,for applications, we need smooth commuting functions that are globally defined on the whole manifold M . For two constant brackets { , } and { , } on a vector space V , bi-integrable systems can be easilyconstructed by means of bi-Poisson linear algebra: each bi-Lagrangian subspace can be interpreted asa bi-integrable Hamiltonian system with linear integrals. The first non-trivial example is the so-called“argument shift” pencil (see A.S. Mischenko and A.T. Fomenko [172]), generated by a linear and aconstant brackets. We recall this construction. 25 .2 Argument shift method and Jordan-Kronecker invariants of Lie alge-bras Let g ∗ be the dual space of a finite-dimensional Lie algebra g . It is well known that g ∗ possesses twonatural compatible Poisson brackets. The first one is the standard linear Lie-Poisson bracket { f, g } ( x ) = h x, [ df ( x ) , dg ( x )] i , (5.3)and the second one is a constant bracket given by { f, g } a ( x ) = h a, [ df ( x ) , dg ( x )] i , (5.4)where a ∈ g ∗ is a fixed element. Here we assume a to be regular, although formula (5.4) makes sensefor an arbitrary a .There are many examples of finite-dimensional Lie algebras for which we can construct a completefamily of polynomials in bi-involution with respect to the brackets (5.3) and (5.4). In particular, thefamous theorem by A.S. Mischenko and A.T. Fomenko [172] states that such a family exists for everysemisimple Lie algebra g and consists of shifts of Ad ∗ -invariant polynomials, i.e., function of the form f i ( x + λa ), where λ ∈ R and f , . . . , f r are generators of the algebra of (co)-adjoint invariants.However, it is still an open question whether one can construct such a family for every finite-dimensionalLie algebra. In many examples (see [38]), the answer turns out to be positive, leading us to the followingGeneralised Argument Shift conjecture which was first stated in [38] and then mentioned and discussedin a number of papers [195, 36, 122, 34]: Conjecture 5.5.
Let g be a finite-dimensional Lie algebra. Then for every regular element a ∈ g ∗ ,there exists a complete family G a of polynomials in bi-involution, i.e., in involution with respect to thetwo brackets { , } and { , } a . It is expected that the solution to this conjecture will essentially rely on the algebraic type of thepencil { , } λ = { , } + λ { , } a or, equivalently, of the pencil of the skew-symmetric forms A x + λa , λ ∈ C , on g for a generic pair ( x, a ) ∈ g ∗ × g ∗ , where A x + λa ( ξ, η ) = h x + λa, [ ξ, η ] i , ξ, η ∈ g . This algebraic type can be considered as an invariant of the Lie algebra g , called Jordan–Kroneckerinvariant (see [38]).
Problem 5.6.
Compute the Jordan–Kronecker invariants for the most interesting classes of Lie alge-bras, and particularly for the following ( a ) semidirect sums g + ρ V , where ρ : g → End( V ) is a representation of a semisimple Lie algebra g and V is assumed to be commutative; ( b ) Borel subalgebras of simple Lie algebras; ( c ) parabolic subalgebras of simple Lie algebras; ( d ) the centralisers of singular elements a ∈ g , where g is simple; ( e ) Lie algebras of small dimensions. ≤
5, the Jordan–Kronecker invariants were computed by P. Zhang. Herresults, as well as some other examples of computing Jordan–Kronecker invariants, can be found inthe arXiv version of [38]. For some semidirect sums, Jordan-Kronecker invariants have been recentlycomputed by K. Vorushilov [224].
Question 5.7.
Are there any restrictions on the algebraic type of the pencils A x + λa ? Which algebraictypes can be realised by means of an appropriately chosen Lie algebra ?A recent observation by I. Kozlov (see details in [38]) shows that some restrictions do exist. Roughlyspeaking, Kozlov has obtained the answer for Lie algebras of pure Jordan and pure Kronecker type.The mixed case remains open because of possible non-trivial interaction between Kronecker and Jordanblocks.To further develop these ideas, it would be interesting to study Poisson pencils generated by onequadratic and one linear brackets. Compatible brackets of this kind appear, in particular, in thetheory of Poisson-Lie groups. Problem 5.8.
Study examples of “quadratic + linear” Poisson pencils. Compute their algebraic typesand construct complete families of polynomials in bi-involution for such pencils. It is a very general question whether or not a given symplectic (Poisson) manifold M admits anintegrable system with good properties. In particular, if M is a vector space endowed with a polynomialPoisson bracket, we are interested in integrable systems defined on M by a collection of Poissoncommuting polynomials. For a linear Poisson bracket (5.3), such a polynomial integrable systemalways exists. This statement, known as Mischenko-Fomenko conjecture, was proved by S. Sadetov[197] in 2004. What can we say about polynomial Poisson brackets of higher degrees? Question 5.9.
Is it true that for any quadratic Poisson bracket (defined on a vector space), thereexists a polynomial integrable system?
Let L = (cid:0) L ij ( x ) (cid:1) be a (1 , M .We say that L is a Nijenhuis operator if its Nijenhuis torsion N L identically vanishes on M , i.e., N L ( v, w ) = L [ v, w ] + [ Lv, Lw ] − L [ Lv, w ] − L [ v, Lw ] = 0 , for arbitrary vector fields v, w on M .It is known that Nijenhuis operators are closely related to compatible Poisson brackets and thereforenaturally appear in the theory of finite dimensional integrable systems, see e.g. Magri et al [140, 154].Indeed, if a Poisson structure given by the tensor P ij is compatible with the Poisson structure givenby the symplectic form ω , then L ij := P is ω sj is a Nijenhuis operator. Nijenhuis tensors appear alsoin other topics in the theory of integrable systems, see e.g. [24, 119], in differential geometry, see e.g.276, 29, 33], so investigation of Nijenhuis tensors is expected to be generally important for the theory ofintegrable systems and other branches of mathematics; in this section we collect open problems aboutit.Let L = L ( x ) be a (1 , x ∈ M , we can characterize L ( x ) by itsalgebraic type (Segre characteristic), that is, the type of its Jordan normal form (eigenvalues withmultiplicities + sizes of Jordan blocks for each eigenvalue). The algebraic type of L depends on x andmay vary from point to point. We will say that x ∈ M if generic , if the algebraic type of L is locallyconstant at x (i.e., is the same for all points from a certain neighbourhood U ( x )). Otherwise, x iscalled singular . One can show that generic points form an open dense subset of M .Almost all results known in the literature concern local properties of Nijenhuis operators at genericpoints, e.g. for semisimple Nijenhuis operators a local description is due to Haantjes [109]. However,Nijenhuis operators also occur in problems which require a global treatment and analysis of singularpoints. Indeed, in the theory of bi-Hamiltonian systems, the Nijenhuis operators appear as recursionoperators R (see [154, 140]). The first integrals (for an important class of integrable systems) arethe coefficients of the characteristic polynomial of R . This means, in particular, that the structure ofsingularities of the corresponding Liouville foliations is determined by the local structure of singularpoints of R . Analysing them, we may find topological obstruction to bi-Hamiltonisation, which wouldbe an interesting result. Philosophically, we want to say that the singularities of bi-integrable systemsare determined by the singularities of the underlying structure (in our case, this is just the recursionoperator). We also notice that integrability of finite-dimensional Hamiltonian systems is essentiallya global phenomenon, so that it might be important to understand global properties of recursionoperators.Similarly, in differential geometry, Nijenhuis operators play an important role in the theory of pro-jectively equivalent (pseudo-)Riemannian metrics, where understanding of singular points of relatedNijenhuis tensors led to the proof of the projective Lichnerowicz conjecture in the Riemannian andLorentzian signature [161, 35].In view of the above discussion, we find the following directions of research very important: • Studying global properties of Nijenhuis operators and topological obstructions to the existenceof Nienhuis operators (of a certain type) on compact manifolds. • Studying local behaviour of Nijenhuis operators at singular points.Speaking of topological obstructions to the existence of Nijenhuis opertors on compact manifolds, weshould take into account two simple but important observations:(i) The operators of the form f ( x ) · Id are Nijenhuis for any smooth function f : M → R . Suchoperators exist on any manifold but this example is too trivial.(ii) In R n , there exist C ∞ -smooth Nijenhuis operators with compact support and such that, in acertain domain, they take the standard diagonal form L ( x ) = diag( λ ( x ) , λ ( x ) , . . . , λ n ( x n ))28ith distinct non-constant real eigenvalues λ i ( x i ). Since the support of such an operator iscontained in a certain ball, we can construct a Nijenhuis operator with similar properties on any(compact) manifold.This leads to the following natural questions.We will say that a linear operator L : V → V is algebraically regular if the dimension of the orbit of L under the canonical GL ( V )-action on the space of operators is maximal (equivalently, the dimensionof the centraliser of L is minimal, i.e., equals n = dim V ). In particular, diagonalisable operatorswith simple spectrum are regular. For non-diagonalisable operators, the algebraic regularity conditionmeans that geometric multiplicity of each eigenvalue of L is one. Problem 5.10.
Describe closed manifolds M which admit Nijenhuis operators L ( x ) that are alge-braically regular at each point x ∈ M . Problem 5.11.
Let us fix a certain algebraic type of a linear operator, i.e., its Segre characteristic(see above). Does there exist a Nijenhuis operator L in R n with the following two properties:1. L has compact support;2. in a certain domain U ⊂ R n , the algebraic type of L does not change and coincides with thegiven one?More generally, find necessary and sufficient conditions, in terms of the Segre characteristic, for theexistence of a Nijenhuis operator in R n with the above properties. Notice that for semisimple operators L with real eigenvalues, the answer to the above question ispositive. However, if some of the eigenvalues of L are complex, then L cannot have compact support.Also, one can easily construct an example of a Nijenhuis operator L with compact support which hasthe form of a Jordan block, e.g., L = (cid:18) y x y (cid:19) , on a certain disc. However, even for single Jordanblocks of size ≥ Problem 5.12.
Construct real analytic examples of Nijenhuis operators on closed two-dimensionalsurfaces whose eigenvalues are real and generically distinct.
Let χ ( t ) = det( t · Id − L ) = t n + σ ( x ) t n − + σ ( x ) t n − + · · · + σ n ( x ) be the characteristic polynomialof a Nijenhuis operator L . Then the following relation holds [29]: JL = − σ − σ − σ n − − σ n J, where J = ∂σ ∂x . . . ∂σ ∂x ... . . . ... ∂σ n ∂x . . . ∂σ n ∂x (5.5)29his formula shows that, in the case when the coefficients of the characteristic polynomial of L arefunctionally independent, a Nijenhuis operator L can be explicitly reconstructed from σ , . . . , σ n .Notice that, by continuity, this is still true if σ , . . . , σ n are independent almost everywhere on L . Inparticular, the singularities of L will be completely determined by the singularities of the smooth mapΦ = ( σ , . . . , σ n ) : M → R n . However, the singularities of Φ must be very special.We can illustrate this with a very simple example. Let dim M = 2, then we only have two coefficients σ and σ . Consider a point where dσ and dσ are linearly dependent and d Φ has rank 1. Assume,in addition, that dσ = 0 so that there exists a local coordinate system ( x, y ) such that σ = x (and σ = f ( x, y )). Then (5.5) implies that, in coordinates ( x, y ), our Nijenhuis operator L takes the form L = 1 f y (cid:18) f y − f x (cid:19) (cid:18) − x − f (cid:19) (cid:18) f x f y (cid:19) = f x − x f yf x ( x − f x ) − ff y − f x ! . Thus, σ = f ( x, y ) is admissible if and only if f x ( x − f x ) − ff y is a smooth function. In other words, todescribe singular points of L in this rather simple particular case we need to solve the following Problem 5.13.
Describe all the functions f ( x, y ) of two variables defined in a neighbourhood of (0 , ∈ R such that • f y (0 , ; • f y (0 ,
0) = 0 ; • f x ( x − f x ) − ff y is locally smooth (or, more precisely, extends up to a locally smooth function). A more general version of this question (which would immediately clarify the local structure of singularpoints for a wide class of Nijenhuis operators) is as follows.
Problem 5.14.
Consider a smooth map
Φ = ( σ , . . . , σ n ) : U (0) → R n , where U (0) is a neighbourhoodof the origin ∈ R n . We assume that is a singular point of Φ , that is, rank d Φ(0) < n , but almostall points x ∈ U (0) are regular. Describe those maps Φ for which all the components of the Nijenhuisoperator L ( x ) defined by (5.5) are smooth functions (notice that L is well defined and smooth at regularpoints of Φ , and we are interested in necessary and sufficient conditions for L to be smoothly extendableonto the whole neighbourhood U (0) ). The next portion of problems is related to Nijenhuis operators all of whose components are linearfunctions in local coordinates x , . . . , x n , that is, L ij ( x ) = X k b ijk x k (5.6)The role of such operators in the theory of Nijenhuis operators is similar to the role of linear Poissonbrackets in Poisson geometry. Recall that linear Poisson brackets are in a natural one-to-one cor-respondence with finite-dimensional Lie algebras. In the context of Nijenhuis operators, we have asimilar statement (see, e.g., [231]). 30 roposition 5.15. An operator L ( x ) with linear entries (5.6) is Nijenhuis if and only if b ijk arestructure constants of a left-symmetric algebra. Recall that an algebra ( A, ∗ ) is called left-symmetric if the following identity holds: a ∗ ( b ∗ c ) − ( a ∗ b ) ∗ c = b ∗ ( a ∗ c ) − ( b ∗ a ) ∗ c, for all a, b, c ∈ A. The definition is due to Vinberg [223]; recent papers on left-symmetric algebras include [43, 44].Only few results are known about left-symmetric algebras. In particular, constructing new examplesand classifying left-symmetric algebras of low dimensions would be a very interesting problem in thecontext of the theory of Nijenhuis operators.For a given left-symmetric algebra with the structure constants b ijk , consider the coefficients σ , . . . , σ n of the characteristic polynomial of L ( x ) = (cid:16) L ij = P k b ijk x k (cid:17) . It is easy to see that σ k is a homogeneouspolynomial of degree k . Consider the class of left-symmetric algebras for which these polynomialsare algebraically independent. From (5.5) we conclude that the classification of such left-symmetricalgebras is equivalent to that of collections of polynomials σ , . . . , σ n , deg σ k = k , satisfying thefollowing rather exceptional property.Consider the Jacobian matrix J = (cid:16) ∂ ( σ ,...,σ n ) ∂ ( x ,...,x n ) (cid:17) . It is easy to see that D = det J is a homogeneouspolynomial of degree n ( n − . Consider the matrix L = J − − σ − σ n − − σ n . . . J, where J = (cid:18) ∂ ( f , . . . , f n ) ∂ ( x , . . . , x n ) (cid:19) It is easy to check that the entries of this matrix are rational functions of the form L ij = Q ij D where Q ij are homogeneous polynomials of degree deg D + 1. In other words, the entries of L are homogeneousrational functions of degree one. For some special polynomials σ , . . . , σ n , a miracle happens: all Q ij turn out to be divisible by D and the entries of L become linear functions in x , . . . , x n . Such L areautomatically Nijenhuis operators (equivalently, LSA’s). Problem 5.16.
Describe/classify the collections of algebraically independent homogeneous polynomials σ , . . . , σ n , deg σ k = k , in n variables x , . . . , x n with the required property. Finally, one question revealing the relationship between singularities of bi-Hamiltonian systems andsingular points of Nijenhuis operators. In a recent paper [31], it was shown that the structure ofsingularities of bi-Hamiltonian systems related to Poisson pencils of Kronecker type is determined bythe singularities of the pencil. We expect that a similar principle works for Poisson pencils of sympletictype, i.e., when one of two compatible Poisson brackets is non-degenerate. In this case, we have awell-defined recursion operator R (which is automatically a Nijenhuis operator) and the commuting31unctions we are interested in are just the coefficients σ , . . . , σ n of the characteristic polynomial of R . We assume that they are independent almost everywhere on M and want to study singular pointsof the mapping Φ = ( σ , . . . , σ n ) : M → R n playing the role of the momentum mapping for ourbi-Hamiltonian system. Problem 5.17.
Describe the structure of singularities of
Φ = ( σ , . . . , σ n ) : M → R n in terms of thesingular points of the recursion operator R , their linearisations, and the corresponding left-symmetricalgebras. What are sufficient and/or necessary conditions for such singularities to be non-degenerate(in the sense of Eliasson [72])? Notice that, in this case, the algebraic type of R is rather special: ata generic point, R is semisimple, but each of its eigenvalues has multiplicity 2. Poisson manifolds constitute a natural generalisation of symplectic manifolds and their Hamiltoniandynamics generalises that of symplectic manifolds and adds singularities to its complexity (see thebeginning of Section 5 for basic definitions).Let Π denote the Poisson tensor (bi-vector) on M defined, as usual, by Π( df, dg ) = { f, g } . A Hamil-tonian vector field in the Poisson category is defined by the following equation X f := Π( df, · ). Theset of Hamiltonian vector fields on a Poisson manifold defines a distribution which is integrable in thesense that we may associate a (singular) foliation to it. This foliation is called the symplectic foliation of the Poisson manifold ( M, Π). The dimension of the symplectic leaf through a point coincides withthe rank of the bivector field Π at the point. The functions defining this foliation (if they exist) arecalled Casimirs. A manifold with constant rank is called a regular
Poisson manifold.The following theorem by Weinstein [230] gives the local picture of this foliation. A Poisson manifold islocally a direct product of a symplectic manifold, endowed with its Darboux symplectic form, togetherwith a transverse Poisson structure whose Poisson vector field vanishes at the point.
Theorem 6.1 (Weinstein splitting theorem) . Let ( M n , Π) be a Poisson manifold of dimension n andlet the rank of Π be k at a point p ∈ M . Then, in a neighbourhood of p , there exists a coordinatesystem ( x , y , . . . , x k , y k , z , . . . , z n − k ) such that the Poisson structure can be written as Π = k X i =1 ∂∂x i ∧ ∂∂y i + n − k X i,j =1 f ij ( z ) ∂∂z i ∧ ∂∂z j , (6.1) where f ij are functions vanishing at the point and depending only on the transversal variables ( z , . . . , z n − k ) . The notion of integrable systems on a Poisson manifold (see also Section 5) stands for a collection ofintegrable systems on this symplectic foliation. Namely, let ( M, Π) be a Poisson manifold of (maximal)rank 2 r and of dimension n . An s -tuplet of Poisson-commuting functions F = ( f , . . . , f s ) on M is said32o define a Liouville integrable system on ( M, Π) if f , . . . , f s are independent (i.e., their differentialsare independent on a dense open set) and r + s = n .The map F : M → R s is often called the moment map of ( M, Π , F ).For integrable systems on Poisson manifolds it is possible to obtain an action-angle theorem in aneighbourhood of a regular torus, as it was proven in [144], showing that, up to a diffeomorphism, theset of integrals F is indeed the moment map of a toric action: Theorem 6.2 (Laurent-Gengoux, Miranda, Vanhaecke) . Let ( M, Π) be a Poisson manifold of dimen-sion n of maximal rank r . Suppose that F = ( f , . . . , f s ) is an integrable system on ( M, Π) and m ∈ M is a point such that(1) df ∧ · · · ∧ df s = 0 ;(2) The rank of Π at m is r ;(3) The integral manifold L m of X f , . . . , X f s , passing through m , is compact.Then there exists R -valued smooth functions ( σ , . . . , σ s ) and R / Z -valued smooth functions ( θ , . . . , θ r ) ,defined in a neighbourhood U of L m , such that1. The manifold L m is a torus T r ;2. The functions ( θ , . . . , θ r , σ , . . . , σ s ) define an isomorphism U ≃ T r × B s ;3. The Poisson structure can be written in terms of these coordinates as Π = r X i =1 ∂∂θ i ∧ ∂∂σ i , in particular, the functions σ r +1 , . . . , σ s are Casimirs of Π (restricted to U );4. The leaves of the surjective submersion F = ( f , . . . , f s ) are given by the projection onto the sec-ond component T r × B s , in particular, the functions σ , . . . , σ s depend on the functions f , . . . , f s only. The functions θ , . . . , θ r are called angle coordinates , the functions σ , . . . , σ r are called action coordi-nates , and the rest of the functions σ r +1 , . . . , σ s are called transverse coordinates .Observe, in particular, that this theorem provides not only the existence of Liouville tori and action-angle coordinates in a neighbourhood of an invariant submanifold of a Poisson manifold, but also itensures that, in a neighborhhod of an integral manifold L m , there is a set of Casimirs. Difficulties arisewhen we try to extend this theorem to points where the Poisson structure is not regular. For particularclasses of Poisson manifolds where some transversality conditions are met (such as b -Poisson manifolds33101, 102]), it has been possible to extend this action-angle coordinates scheme (see [134, 135] and also[107]).The first open problem concerns this extension: Problem 6.3.
Extend the action-angle theorem in a neighbourhood of a Liouville torus at singularpoints of the Poisson structure.
The first adversity in doing so is that, contrary to the expectations, in general, an integrable system isnot a product of two integrable systems (one on the symplectic leaf and the other one in the transversepart). So there is no simultaneous splitting theorem for the Poisson structure and the integrablesystem. Counterexamples to this decomposability can be found in [143]. When this happens, we saythe system is splittable. So the problem above can be refined as follows:
Problem 6.4.
Extend the action-angle theorem in a neighbourhood of a Liouville torus at singularpoints of the Poisson structure for splittable integrable systems.
We suspect that either stability of the transversal part of the Poisson manifold or the existence ofassociated toric actions will play a role in solving this problem for special classes of Poisson manifolds.This problem can be extended to its global version. Can we find action-angle coordinates globally?This problem was considered by Duistermaat in the symplectic case [68]. One of the main ingredientsin his construction was the affine structure defined by the action variables on the image of the momentmap. In the regular Poisson setting, the image of the moment map is foliated by affine manifolds.Understanding this foliation is a key point to understanding global obstructions. The following problemhas been partially considered in [76].
Problem 6.5.
Determine obstructions to the global existence of action-angle coordinates on regularPoisson manifolds.
Realisation problem is also important:
Problem 6.6.
Which foliations with affine leaves can be described as the image of the moment map?
Notice that a similar problem is still open even in the symplectic setting: which closed manifolds B n may serve as bases of locally trivial Lagrangian fibrations M n T n −→ B n , where M n is symplectic?Extensions of the affine structure to the singular setting are also desirable. One instance when theaffine structure on the leaves extends to an affine structure on the image of the moment map fornon-regular Poisson manifolds is the case of b -Poisson manifolds [107]. For these manifolds this affinestructure is well-understood and classified via the Delzant theorem [106]. Such a phenomenon is alsoobserved for other Poisson structures with open dense leaves on b m -Poisson manifolds [104]. Problem 6.7.
Consider a Poisson manifold M of even dimension such that it is symplectic on adense set U ⊂ M . Assume that M is endowed with a toric action which is Hamiltonian on U . Is therean analog of Delzant theorem for the image of the moment map? • The Chern class always vanishes for two degrees of freedom systems; • There are no known physically meaningful examples with non-trivial Chern class in three degreesof freedom.Although it is easy to manually construct any Chern class, see [8], having a natural example fromclassical mechanics would provide better motivation for further study (see also Problem 7.4).
Problem 6.8 (S. V˜u Ngo. c) . Find natural examples of integrable systems in classical mechanics withnon-trivial Duistermaat–Chern classes.
The following open problems concern the quantum aspect of Liouville integrable systems. From thequantum mechanical perspective, the most natural notion of finite dimensional quantum integrablesystems is obtained by replacing classical observables (smooth real functions on a symplectic manifold)by quantum observables (self-adjoint operators on a Hilbert space), and, due to the correspondenceprinciple of Dirac, Poisson brackets by commutators.Therefore, a quantum integrable system on a “quantum Hilbert space corresponding to M n ” shouldbe the data of n semiclassical operators P , . . . , P n such that1. P ∗ j = P j
2. [ P i , P j ] = 03. the P j ’s are “independent” (see below, and Problem 7.2).The word semiclassical is very important and means that such operators P j should have a “classicallimit”, typically in the regime when Planck’s constant ~ tends to zero. Without this semiclassical limit,the number n , which relates to the dimension 2 n of the classical phase space, would be irrelevant.35here are several instances of quantum settings when the semiclassical limit can be made precise. Thetwo prominent theories are pseudo-differential operators (when M is a cotangent bundle) and Berezin-Toeplitz operators (when M is a prequantisable symplectic manifold). In both cases, operators P havea principal symbol p = σ ( P ), which is a smooth function on M , and Dirac’s correspondence principleholds modulo O ( ~ ), which implies that [ P j , P k ] = 0 = ⇒ { p j , p k } = 0. We can now explain thelast item above: we say that the P j ’s are independent when the principal symbols p j ’s have (almosteverywhere) independent differentials.The following very natural and classical question that one can ask immediately after defining quantumintegrable systems is still open and probably quite difficult. Problem 7.1.
Assume that a classical integrable system ( p , . . . , p n ) is given on M . Does there exista quantum integrable system ( P , . . . , P n ) such that p j = σ ( P j ) ? See Charles et al [49] for a solution in the compact toric case. More specific versions of this generalquestion are discussed in Section 7.3.The difficulty is to construct operators P j with the exact commutation property [ P j , P k ] = 0. Us-ing microlocal analysis and, in particular, Fourier integral operators [66], one should be able toobtain [ P j , P k ] = O ( ~ ∞ ). In many classical examples, an exact quantisation is known, see for in-stance [215, 114]. Symmetry considerations should help greatly. This is a typical local-to-globalobstruction problem, and one expects a corresponding cohomology complex. In the case of polynomialsymbols in C n , the complex was constructed in [78]. Problem 7.2 (S. V˜u Ngo. c) . Given a set of semiclassical operators that verify conditions 1 and 2above, can one detect the independence axiom 3 in a purely spectral way? More precisely, can one tellfrom asymptotics of joint eigenvalues or eigenfunctions of ( P , . . . , P n ) that the principal symbols arealmost everywhere independent? A great achievement of microlocal analysis of pseudo-differential operators is the formulation of com-plete Bohr-Sommerfeld rules to any order . Given a quantum integrable system, these rules are, in fact,some kind of integrality conditions (or cohomological conditions), depending on a parameter E ∈ R n ,that are satisfied if and only if E is a joint eigenvalue of the system, modulo an arbitrarily small errorof size O ( ~ ∞ ).Historically (both in physics and mathematics), these rules have been written for regular Lagrangiantori, i.e., E should stay in a small ball of regular values of the classical moment map. More recently,non-trivial extensions to some singular situations have been found, see [50, 229, 228].In order to treat even more physical examples, and to appeal to the geometry community, it is impor-tant to extend these rules to Berezin-Toeplitz quantisation on compact K¨ahler manifolds. This wasdone recently for regular tori [48], and for one degree of freedom systems with Morse singularities,see [147, 146]. It should be possible to treat the focus-focus case as well, along the lines of [227]. Problem 7.3 (S. V˜u Ngo. c) . Write Bohr-Sommerfeld rules for focus-focus singularities in Berezin-Toeplitz quantisation.
36 similar problem appears in the context of geometric quantisation (see Problem 7.15 below).In relation to Problem 6.8, the following natural question is still open: what is the quantum analogueof the Chern (or Duistermaat-Chern) class? Contrary to Quantum Monodromy, which was definedpurely in terms of the joint spectrum [56, 226], it is quite possible that the Chern class manifestsitself as a phase on eigenfunctions, and thus could be invisible in the spectrum. This study probablyinvolves “shift operators” that are transversal to the Lagrangian fibration and allow to “jump” fromone eigenspace to another one.
Problem 7.4 (S. V˜u Ngo. c) . Define (and detect) the quantum Chern class.
The notion of quantum integrable system as a maximal set of commuting quantum observables is fairlywell established now. It is a very old notion, going back to Bohr, Sommerfeld and Einstein [71] in theearly days of quantum mechanics. Nonetheless the theory didn’t blossom at that time because of thelack of technical tools needed to solve interesting problems about quantum integrable systems. Forrecent surveys discussing the spectral geometry of integrable systems, see [188, 181].The most elementary results about the symplectic structure of classical integrable systems, such asthe existence of action-angle variable, did not fit well in Schr¨odinger’s quantum setting at that timebecause they involve the analysis of differential and pseudodifferential operators, only developed sincethe 1960s. This is a prominent part of analysis and PDE (the most mathematical side of quantummechanics), that goes by the name of “microlocal analysis”.The microlocal analysis of action-angle variables starts with the pioneering works of Duistermaat [67]and Colin de Verdi`ere [51, 52], and was followed by many other authors (see [49] for further references).For compact symplectic manifolds, the study of quantum action-angle variables is very recent, it goesback to Charles [48], using the theory of Toeplitz operators. Toeplitz operators give rise to a semiclas-sical algebra of operators with symbolic calculus and microlocalization properties, isomorphic (at amicrolocal level) to the algebra of pseudodifferential operators (see Boutet de Monvel-Guillemin [42]).The classification of semitoric systems in [185, 186] (see Section 1) goes a long way to answering thefascinating question “can one hear the shape of a semitoric system”? More precisely, the idea is torecover the classical semitoric invariants from the joint spectrum of a quantum integrable system, asmuch as possible.
Conjecture 7.5 (Pelayo-V˜u Ngo. c 2011, Conjecture 9.1 in [187]) . A semitoric system F = ( J, H ) is determined, up to symplectic equivalence, by its semiclassical joint spectrum (i.e., the collectionof points ( λ, ν ) ∈ R , where λ is a eigenvalue of ˆ J ~ and ν is an eigenvalue of ˆ H ~ , restricted to the λ -eigenspace of ˆ J , as ~ → ). Moreover, from any such semiclassical spectrum, one can explicitlyconstruct the associated semitoric system. This conjecture is now a theorem for the special cases of quantum toric integrable systems on compact37anifolds [49], and for semitoric integrable systems of Jaynes-Cummings type [149]. The problem isthat it is unclear whether one of the most subtle (perhaps the most subtle) invariants of integrablesystems, known as the twisting index, can be detected in the semiclassical spectral asymptotics. If thisis the case (which is very likely), then the Pelayo-V˜u Ngo. c inverse spectral conjecture ([187]) abovewill hold.An important part of the problem is to recover the Taylor series invariant that classifies semi-globalneighbourhood s of focus-focus fibers [225]. A positive solution was given in [189, 149], but theprocedure was not constructive: one had to first let ~ → c near the focus-focusvalue c , and then take the limit c → c , which doesn’t help computing the Taylor series in an explicitway from the spectrum. Problem 7.6 (S. V˜u Ngo. c) . Compute the Taylor series invariant of semitoric systems directly fromthe spectrum “at” the focus-focus critical value.
One should probably use the corresponding singular Bohr-Sommerfeld rule, see Problem 7.3. In fact,the following intermediate question is very natural:
Problem 7.7 (S. V˜u Ngo. c) . Is the singular Bohr-Sommerfeld formal power series in ~ a spectralinvariant? There are various later refinements and extensions of Conjecture 7.5 [188, 190, 198], following otherinverse spectral questions in semiclassical analysis (see for instance [234, 115, 118]) which concernintegrable systems (or even collections of commuting operators) more general than semitoric, and evenin higher dimensions. All of them essentially make the same general claim: “from the semiclassicaljoint spectrum of a quantum integrable system one can detect the principal symbols of the system”.A more concrete problem is:
Problem 7.8 ( ´A. Pelayo) . What information about the principal symbols f , . . . , f n of a quantumintegrable system T , ~ , . . . , T n, ~ can be detected from their semiclasical joint spectrum? For instance,suppose that we know that a certain object – say, an integer z , or a matrix A – is a symplectic invariantof ( M, ω, f , . . . , f n ) . Can we compute this object from the semiclassical joint spectrum? An additional interesting problem, which is in fact closely related to Question 2.5 is as follows:
Problem 7.9 ( ´A. Pelayo) . Can one detect from the joint spectrum of a quantum integrable system T , ~ , . . . , T n, ~ that a singularity is degenerate? The theory of semitoric systems provides a strong motivation for understanding the connections be-tween integrable systems and Hamiltonian torus actions. In view of this, we conclude by discussing apossible quantum approach to counting fixed points of symplectic S -actions.In view of Tolman’s recent work constructing a non-Hamiltonian symplectic S -action on a compactmanifold with isolated fixed points, the questions how many fixed points symplectic actions have (bothupper and lower bounds) and what the relation between having fixed points and being Hamiltonian38s, have attracted much interest [214, 92, 184, 93]. See also Pelayo [180] for a discussion and otherreferences. As far as we know, the “quantised” version of this problem is not well understood, althoughsome steps have been taken in this direction [148]. The object to quantise can no longer be themomentum map, since there is no momentum map — however, McDuff pointed out in [164] that anysymplectic S -action admits an S -valued momentum map µ : M → S . The explicit construction ofthis map appears in [182], and it can be quantised [148, Section 7] (here the operator will be unitary,rather than self-adjoint). Problem 7.10 ( ´A. Pelayo) . Can one make progress in counting the number of fixed points by studyingthe spectrum of the quantisation of µ : M → S ? This type of problem should be very relevant to the study of equilibrium points of integrable systemsbut, for the moment, it is only a toy version to start with. The general case would involve quantisationand a study of the fixed point properties of an object (momentum map) of the form M → ( S ) k × R n ,and this will likely be quite challenging. The approach to quantisation of natural Hamiltonian systems discussed in this section is, perhaps,the most traditional. Instead of a Hamiltonian of the form (3.1), i.e., H = K + U on a Riemannianmanifold M , one considers the second order differential operator acting on C -differentiable functionson M and given by ˆ H := ∆ + U, where ∆ = g ij ∇ i ∇ j is the Beltrami-Laplace operator corresponding to g . Next, by a quantum integralwe understand another differential operator ˆ F that commutes with H :[ ˆ H, ˆ F ] = ˆ H ◦ ˆ F − ˆ F ◦ ˆ H = 0 . (7.1)Sometimes one requires that the operator ˆ F be self-adjoint. It is known and is easy to see that, for anyoperators ˆ H and ˆ F , the symbol of [ ˆ H, ˆ F ] is just the Poisson bracket of the symbols of ˆ H and ˆ F . Wesay that a polynomial in momenta integral F for (3.1) is quantisable if there exists a quantum integralˆ F for ˆ H such that its symbol is F . Problem 7.11.
What are necessary and/or sufficient conditions on a metric g such that every poly-nomial integral of its geodesic flow is quantisable? Partial answers are known in the case of integrals of small degrees and in dimension 2.A more specific version of Problem 7.11 is related to quantisation of the argument shift method (seeSection 5.2). In this setting, we impose an additional condition on classical and quantum integrals.Namely, we assume that M = G is a Lie group, and the polynomial commuting integrals are allleft invariant. Is it possible to find the corresponding commuting quantum integrals, i.e., differential39perators, that are still left invariant? More specifically, this question is related to a special family(algebra) of left invariant polynomial integrals on T ∗ G obtained by the argument shift method [172],and it can be asked in a purely algebraic terms.Let P ( g ) be the algebra of polynomials on g ∗ endowed with the standard Lie-Poisson bracket (5.3),and a ∈ g ∗ be a regular element. For a local Ad ∗ -invariant function f (not necessarily polynomial),consider its Taylor expansion at the point a ∈ g ∗ : f ( a + λx ) = f ( a ) + λf ( x ) + λ f ( x ) + · · · + λ k f k ( x ) + . . . with f k ( x ) ∈ P ( g ) being a homogeneous polynomial of degree k . The subalgebra F a ⊂ P ( g ), generatedby f k ’s (where k ∈ N and f runs over the set of local Ad ∗ -invariant functions), is commutative withrespect to the Lie-Poisson bracket (5.3) and is called a Mischenko-Fomenko subalgebra of P ( g ). Eachpolynomial g ∈ F a can be naturally lifted to the cotangent bundle T ∗ G and, as a result, we obtain acommutative subalgebra of left-invariant functions, polynomial in momenta on T ∗ G . Quantising thisalgebra in the class of left-invariant differential operators is equivalent to “lifting” F a to the universalenveloping algebra U ( g ), i.e., constructing a commutative subalgebra b F a ⊂ U ( g ) such that for anyelement ˆ p ∈ b F a of degree k , its “principal symbol” p ∈ P ( g ) (being a homogeneous polynomial ofdegree k ) lies in the algebra F a , and F a is generated by such elements.A method for constructing the quantum Mischenko-Fomenko algebra b F a for a semisimple Lie algebra g was suggested by L.G. Rybnikov [196]. However, it is not clear whether a similar construction worksfor an arbitrary Lie algebra g . Problem 7.12.
Quantise the Mischenko-Fomenko algebra F a ⊂ P ( g ) for an arbitrary finite-dimensionalLie algebra g ( or find an obstruction to quantisation ) . As explained above, the idea of quantisation is to associate a representation space with a classicalsystem in a functorial way, so that the observables (smooth functions) become operators on a Hilbertspace, and the classical Poisson bracket becomes the commutator of operators. Here we would like tostress the role of Lagrangian foliations in Geometric Quantisation.Consider a symplectic manifold ( M n , ω ) with an integral homology class [ ω ]. Because of integralityof [ ω ] ([233], [136]), there exists a complex line bundle L with a connection ∇ over M such that curv ( ∇ ) = ω . The pair ( L , ∇ ) is called a prequantum line bundle of ( M n , ω ). Having fixed these data,as a first candidate for the representation space, we consider the set of sections s which are flat, meaningthat ∇ X s = 0 in some privileged directions, given by polarisation (integrable Lagrangian distributionof the complexified tangent bundle). Any Lagrangian foliation of M n defines a polarisation, and thusintegrable systems can be seen as real polarisations, and their tangent directions can be used to solvethe flatness equation ∇ X s = 0.It is convenient to extend the notion of polarisation to take singularities into account. Thus a realpolarisation P is a foliation whose generic leaves are Lagrangian submanifolds.40 leaf of a polarisation is called Bohr-Sommerfeld if the flat sections are defined along the leaf. Inthe classical case, when M n is a cotangent bundle with the leaves being the cotangent spaces, thiscondition is fulfilled for all of them. However, if the leaves of a polarisation are compact, then thiscondition becomes nontrivial.As we will see next, the action-angle coordinates of integrable systems are useful for localisation ofBohr-Sommerfeld leaves, and a simple model of quantisation would help to count them. This coincideswith Kostant’s viewpoint on geometric quantisation (see [141]).As the simplest example, consider the surface M = S × R endowed with the symplectic form ω = dt ∧ dθ . Take, as a prequantum line bundle L , the trivial bundle with connection 1-form Θ = tdθ ,and as a distribution P = h ∂∂θ i . The flat sections satisfy ∇ X σ = X ( σ ) − i h θ, X i σ = 0.Thus σ ( t, θ ) = a ( t ) e itθ , and Bohr-Sommerfeld leaves are given by the condition t = 2 πk, k ∈ Z , i.e., bythe integral action coordinates, associated to the polarisation with the moment map t .This characterisation of Bohr-Sommerfeld leaves generalises for regular fibrations: Theorem 7.13 (Guillemin-Sternberg, [100]) . Assume the polarisation is a regular Lagrangian fibrationwith compact leaves over a simply connected base B . Then the Bohr-Sommerfeld set is discrete; andif the zero-fiber { ( f ( p ) , . . . , f n ( p )) = 0 ∈ R n } ⊂ M n is a Bohr-Sommerfeld leaf, then the Bohr-Sommerfeld set is given by BS = { p ∈ M, ( f ( p ) , . . . , f n ( p )) ∈ Z n } , where f , . . . , f n are global actioncoordinates on B . Let us now relate Bohr-Sommerfeld leaves with the geometric quantisation scheme. Denote by J thesheaf of (local) flat sections along the polarisation. Following Kostant [141], we define the quantisationas the sheaf cohomology Q ( M ) = L k ≥ H k ( M, J ). The following theorem by Sniatycki [202] providesa bridge between this sheaf cohomology approach of Kostant and counting Bohr-Sommerfeld leaves: Theorem 7.14 (Sniatycki) . If the leaf space B n is Haussdorf and the natural projection π : M n → B n is a fibration with compact fibers, then all the cohomology groups H k ( M, J ) vanish, except for k = dim M . Thus, Q ( M n ) = H n ( M n , J ) , and the dimension of the vector space H n ( M n , J ) equals the number of Bohr-Sommerfeld leaves. This scheme works well for toric manifolds, where the base B may be identified with the image of themoment map associated with the torus action, i.e., the interior of the Delzant polytope and, more gen-erally, for Lagrangian fibrations with elliptic singularities [111]. The next step would be to generalisethis scheme to the case of integrable systems with arbitrary non-degenerate singularities. Unfortu-nately, hyperbolic and focus-focus singularities [112], [167] do not behave well under this scheme, andthe dimension of H n ( M n , J ) may become infinite even if the number of Bohr-Sommerfeld leaves isfinite. Problem 7.15 (Miranda-Presas-Solha [167]) . Modify this scheme to get finite dimensional repre-sentation spaces for focus-focus and hyperbolic singularities that still capture the invariants of thecorresponding integrable systems and their singularities. eferences [1] S. V. Agapov, D. N. Aleksandrov. Fourth-degree polynomial integrals of a natural mechanicalsystem on a two-dimensional torus.
Mat. Zametki (2013), 790–793, English translation Math.Notes (2013), 780–783.[2] P. Albers, S. Tabachnikov. Introducing symplectic billiards. arXiv:1708.07395 .[3] M. Arnold, M. Bialy.
Non-smooth convex caustics for Birkhoff billiard. arXiv:1708.04280 .[4] S. Arthamonov.
Noncommutative inverse scattering method for the Kontsevich system.
Lett. Math.Phys. (2015), 1223–1251.[5] A. Avila, J. De Simoi, V. Kaloshin.
An integrable deformation of an ellipse of small eccentricityis an ellipse , Ann. of Math. (2016), 527–558.[6] P. de Bartolomeis, V. Matveev.
Some Remarks on Nijenhuis Bracket, Formality, and K¨ahlerManifolds.
Adv. Geom. (2013), 571–581.[7] Yu. Baryshnikov, V. Zharnitsky. Sub-Riemannian geometry and periodic orbits in classical bil-liards . Math. Res. Lett., (2006), 587–598.[8] L. Bates. Examples for obstructions to action-angle coordinates.
Proc. Royal Soc. Edinburgh, (1988), 27–30.[9] A. Berenstein, V. Retakh.
A short proof of Kontsevich’s cluster conjecture . C. R. Math. Acad.Sci. Paris (2011), 119–122.[10] M. Berger.
Seules les quadriques admettent des caustiques.
Bull. Soc. Math. France (1995),107–116.[11] A. L. Besse.
Manifolds all of whose geodesics are closed. With appendices by D. B. A. Epstein,J.-P. Bourguignon, L. Brard-Bergery, M. Berger and J. L. Kazdan.
Springer-Verlag, Berlin-NewYork, 1978.[12] M. Bialy.
First integrals that are polynomial in the momenta for a mechanical system on thetwo-dimensional torus.
Funktsional. Anal. i Prilozhen. (1987), no. 4, 64–65.[13] M. Bialy. Convex billiards and a theorem by E. Hopf.
Math. Z. (1993), 147–154.[14] M. Bialy.
Integrable geodesic flows on surfaces.
Geom. Funct. Anal. (2010), 357–367.[15] M. Bialy. On totally integrable magnetic billiards on constant curvature surface.
Electron. Res.Announc. Math. Sci. (2012), 112–119.[16] M. Bialy. Hopf rigidity for convex billiards on the hemisphere and hyperbolic plane.
Discrete Con-tin. Dyn. Syst. (2013), 3903–3913.[17] M. Bialy, A. Mironov. Algebraic non-integrability of magnetic billiards.
J. Phys. A (2016), no.45, 455101, 18 pp. 4218] M. Bialy, A. Mironov. Angular billiard and algebraic Birkhoff conjecture.
Adv. Math. (2017),102–126.[19] P. Blagojevic, M. Harrison, S. Tabachnikov, G. Ziegler.
Counting periodic trajectories of Finslerbilliards. arXiv:1712.07930 .[20] A.V. Bolsinov.
Compatible Poisson brackets on Lie algebras and completeness of families of func-tions in involution.
Math. USSR Izvestiya, (1992), 69–90.[21] A. V. Bolsinov, V. V. Kozlov, A. T. Fomenko. The Maupertuis’ principle and geodesic flowson S arising from integrable cases in the dynamics of rigid body motion. Russ. Math. Surv. (1995), 473–501.[22] A. V. Bolsinov, V. S. Matveev, A. T. Fomenko. Two-dimensional Riemannian metrics with anintegrable geodesic flow. Local and global geometries.
Sb. Math. (1998), no. 9-10, 1441–1466.[23] A. V. Bolsinov, V. S. Matveev.
Singularities of momentum maps of integrable Hamiltonian sys-tems with two degrees of freedom.
Zap. Nauchn. Semin. POMI (1996), 54–86. English trans-lation in J. Math. Sci., New York (1999), No.4, 1477–1500.[24] A. Bolsinov, V. S. Matveev. Geometrical interpretation of Benenti systems.
J. Geom. Phys. (2003), 489–506.[25] A.V. Bolsinov, A.T. Fomenko. Integrable Hamiltonian systems. Geometry, topology, classification.
Chapman & Hall/CRC, Boca Raton, FL, 2004.[26] A. V. Bolsinov, B. Jovanovic.
Integrable geodesic flows on Riemannian manifolds: constructionand obstructions.
Contemporary geometry and related topics, 57–103, World Sci. Publ., RiverEdge, NJ, 2004.[27] A.V. Bolsinov, A.A. Oshemkov.
Bi-Hamiltonian structures and singularities of integrable systems,
Regul. Chaotic Dyn. (2009), 325–348.[28] A. Bolsinov, V. Matveev, G. Pucacco. Normal forms for pseudo-Riemannian 2-dimensional met-rics whose geodesic flows admit integrals quadratic in momenta . J. Geom. Phys. (2009), 1048–1062.[29] A. V. Bolsinov, V. S. Matveev. Splitting and gluing lemmas for geodesically equivalent pseudo-Riemannian metrics.
Trans. Amer. Math. Soc. (2011), 4081–4107.[30] A.V. Bolsinov, A.M. Izosimov, A.Yu. Konyaev, A.A. Oshemkov.
Algebra and topology of integrablesystems. Research problems. (Russian) Trudy Sem. Vektor. Tenzor. Anal. (2012), 119–191.[31] A.V. Bolsinov, A. Izosimov. Singularities of bi-Hamiltonian systems.
Comm. Math. Phys. (2014), 507–543.[32] A.V. Bolsinov, A. V. Borisov, I. S. Mamaev.
Geometrisation of Chaplygin’s reducing multipliertheorem.
Nonlinearity (2015), 2307–2318. 4333] A.V. Bolsinov, V. S. Matveev. Local normal forms for geodesically equivalent pseudo-Riemannianmetrics.
Trans. Amer. Math. Soc. (2015), 6719–6749.[34] A.V. Bolsinov.
Complete commutative subalgebras in polynomial Poisson algebras: a proof of theMischenko-Fomenko conjecture.
Theor. Appli. Mechanics (2016), 145–168.[35] A.V. Bolsinov, V.S. Matveev, S. Rosemann. Local normal forms for c-projectively equivalent met-rics and proof of the Yano-Obata conjecture in arbitrary signature. Proof of the projective Lich-nerowicz conjecture for Lorentzian metrics. arXiv:1510.00275 .[36] A.V. Bolsinov, A.M. Izosimov, D.M. Tsonev.
Finite-dimensional integrable systems: A collectionof research problems.
J. Geom. Phys. (2017), 2–15.[37] A.V. Bolsinov and A.M. Izosimov,
Smooth invariants of focus-focus singularities and obstructionsto product decomposition , arXiv:1706.07456.[38] A.V. Bolsinov, P. Zhang.
Jordan–Kronecker invariants of finite-dimensional Lie algebras.
Trans-form. Groups (2016), 51–86.[39] A.V. Bolsinov, L. Guglielmi, E. Kudryavtseva. Symplectic invariants for parabolic orbits and cuspsingularities of integrable systems with two degrees of freedom. arxiv.org/abs/1802.09910 .[40] D. Bouloc.
Singular fibers of the bending flows on the moduli space of 3d polygons. arXiv:1505.04748 .[41] D. Bouloc, E. Miranda, N.T. Zung.
Singular fibers of the Gelfand-Cetlin system on u ( n ) ∗ . arXiv:1803.08332 .[42] L. Boutet de Monvel, V. Guillemin. The Spectral Theory of Toeplitz Operators . Annals of Mathe-matics Studies, 99. Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo,1981.[43] D. Burde.
Left-symmetric algebras, or pre-Lie algebras in geometry and physics . Central Europ.J. Math. (2006), 323–357.[44] D. Burde. Simple left-symmetric algebras with solvable Lie algebra . Manuscripta Math. (1998),397–411.[45] K. Burns, V. S. Matveev. Open problems and questions about geodesics. arXiv:1308.5417 .[46] L. Butler.
Topology and Integrability in Lagrangian Mechanics.
Chapter 3 in ‘Lagrangian Me-chanics’, Ed. H. Canbolat, 2017, DOI: 10.5772/66147. .[47] R. Casta˜no Bernard, D. Matessi.
Lagrangian 3-torus fibrations.
J. Diff. Geom. (2009), 483–573.[48] L. Charles. Quasimodes and Bohr-Sommerfeld conditions for the Toeplitz operators.
Comm. Par-tial Diff. Equations (2003), 1527–1566. 4449] L. Charles, ´A. Pelayo, S. V˜u Ngo. c. Isospectrality for quantum toric integrable systems.
Ann. Sci.´Ec. Norm. Sup´er. (2013), 815–849.[50] M. S. Child. Semiclassical Mechanics with Molecular Applications.
Oxford University Press, 1991.[51] Y. Colin de Verdi`ere.
Spectre conjoint d’op´erateurs pseudo-diff´erentiels qui commutent. II. Le casint´egrable.
Math. Z. (1980), 51–73.[52] Y. Colin de Verdi`ere.
Spectre conjoint d’op´erateurs pseudo-diff´erentiels qui commutent. I. Le casnon int´egrable.
Duke Math. J. (1979), 169–182.[53] Y. Colin de Verdi`ere, J. Vey. Le lemme de Morse isochore.
Topology (1979), 283–293.[54] S. Console, C. Olmos. Curvature invariants, Killing vector fields, connections and cohomogeneity.
Proc. Amer. Math. Soc. (2008), 1069–1072.[55] L. Corsi, V. Kaloshin.
A locally integrable non-separable analytic geodesic flow. arXiv:1803.01222 .[56] R. Cushman, J. J. Duistermaat.
The quantum spherical pendulum.
Bull. Amer. Math. Soc. (1988), 475–479.[57] G. Darboux, Le¸cons sur la th´eorie g´en´erale des surfaces , Vol. III. Reprint of the 1894 original.´Editions Jacques Gabay, Sceaux, 1993.[58] N. V. Denisova, V. V. Kozlov.
Polynomial integrals of geodesic flows on a two-dimensional torus.
Mat. Sb. (1994), no. 12, 49–64; translation in Russian Acad. Sci. Sb. Math. (1995),469–481.[59] N. V. Denisova, V. V. Kozlov, D. V. Treshchev. Remarks on polynomial integrals of higher degreefor reversible systems with a toral configuration space.
Izv. Ross. Akad. Nauk Ser. Mat. (2012),no. 5, 57–72; translation in Izv. Math. (2012), 907–921.[60] T. Delzant. Hamiltoniens p´eriodiques et images convexe de l’application moment.
Bull. Soc. Math.France, (1988), 315–339.[61] P. Di Francesco, R. Kedem.
Discrete noncommutative integrability: proof of a conjecture by M.Kontsevich . Int. Math. Res. Not. IMRN (2010), 4042–4063.[62] P. Di Francesco, R. Kedem.
Noncommutative integrability, paths and quasi-determinants . Adv.Math. (2011), 97–152.[63] F. Dogru, S. Tabachnikov.
Dual billiards.
Math. Intelligencer (2005), no. 4, 18–25.[64] V. Dragovi´c, M. Radnovi´c. Poncelet porisms and beyond. Integrable billiards, hyperelliptic Jaco-bians and pencils of quadrics.
Frontiers in Mathematics. Birkh¨auser/Springer Basel AG, Basel,2011.[65] J.-P. Dufour, P. Molino, A. Toulet.
Classification des syst`ems int´egrables en dimension 2 et in-variants des mod`eles de Fomenko.
Compt. Rend. Acad. Sci. Paris, (1994), 942–952.4566] J. J. Duistermaat, L. H¨ormander.
Fourier integral operators II.
Acta Math., (1972), 183–269.[67] J.J. Duistermaat.
Oscillatory integrals, Lagrange immersions and unfoldings of singularities.
Comm. Pure Appl. Math. (1974), 207–281.[68] J.J Duistermaat. On global action-angle coordinates , Comm. Pure Applied Math., , (1980),687-706.[69] H. R. Dullin, V. S. Matveev. New integrable system on the sphere.
Math. Res. Lett., (2004),715–722.[70] H. R. Dullin, ´A. Pelayo. Generating hyperbolic singularities in completely integrable systems.
Jour-nal of Nonlinear Science, (2016), 787–811.[71] A. Einstein. Zum Quantensatz von Sommerfeld und Epstein.
Deutsche Physikalische Gesellschaft.Verhandlungen, (1917), 82–92.[72] L.H. Eliasson. Normal forms for Hamiltonian systems with Poisson commuting integrals – ellipticcase.
Comm. Math. Helv. (1990), 4–35.[73] P. Etingof, I. Gelfand, V. Retakh. Nonabelian integrable systems, quasideterminants, andMarchenko lemma.
Math. Res. Lett. (1998), 1–12.[74] V. Fock, A. Marshakov. Loop groups, clusters, dimers and integrable systems.
Geometry andquantization of moduli spaces, 1–66, Adv. Courses Math. CRM Barcelona, Birkh¨auser/Springer,Cham, 2016.[75] R. Felipe, G. Mari Beffa.
The Pentagram map on Grassmannians . To appear in Annales del’Institut Fourier, arXiv:1507.04765 .[76] R. Fernandes, C. Laurent-Gengoux, P. Vanhaecke.
Global action-angle variables for noncommu-tative integrable systems. arXiv:1503.00084 .[77] P. Foth, Y. Hu.
Toric degenerations of weight varieties and applications.
Travaux mathmatiques.Fasc. XVI, 87–105, Trav. Math., 16, Univ. Luxemb., Luxembourg, 2005.[78] M. Garay, D. van Straten.
Classical and quantum integrability.
Mosc. Math. J. (2010), 519–545.[79] M. Gekhtman, M. Shapiro, S. Tabachnikov, A. Vainshtein. Integrable cluster dynamics of directednetworks and pentagram maps.
Adv. Math. (2016), 390–450.[80] I. Gelfand, V. Retakh.
Determinants of matrices over noncommutative rings . Funct. Anal. Appl. (1991), 91–102.[81] I. Gelfand, V. Retakh. Quasideterminants. I.
Selecta Math. (1997), 517–546.[82] I. Gelfand, S. Gelfand, V. Retakh, R. Wilson. Quasideterminants . Adv. Math. (2005), 56–141.[83] I. Gelfand, I. Zakharevich.
Webs, Veronese curves, and bi-hamiltonian systems.
J. Funct. Anal. (1991), 150–178. 4684] D. Genin, S. Tabachnikov. On configuration spaces of plane polygons, sub-Riemannian geometryand periodic orbits of outer billiards . J. Modern Dynamics (2007), 155–173.[85] A. Giacobbe. Infinitesimally stable and unstable singularities of 2-degrees of freedom completelyintegrable systems.
Regul. Chaotic Dyn., (2007), 6, 717–731.[86] P. Gilkey. The Geometry of Curvature Homogeneous Pseudo-Riemannian Manifolds.
ImperialCollege Press Adv. Texts in Math., World Scientific (2007).[87] M. Glick.
The pentagram map and Y -patterns. Adv. Math. (2011), 1019–1045.[88] M. Glick, P. Pylyavskyy. Y -meshes and generalized pentagram maps. Proc. Lond. Math. Soc. (2016), 753–797.[89] A. Glutsyuk.
On 4-reflective complex analytic planar billiards.
J. Geom. Anal. (2017), 183–238.[90] A. Glutsyuk. On polynomially integrable Birkhoff billiards on surfaces of constant curvature. arXiv:1706.04030 .[91] A. Glutsyuk, E. Shustin.
On polynomially integrable planar outer billiards and curves with sym-metry property. arXiv:1607.07593 .[92] L. Godinho.
On certain symplectic circle actions.
J. Symplectic Geom. (2005), 357–383.[93] L. Godinho, ´A. Pelayo, S. Sabatini. Fermat and the number of fixed points of periodic flows.
Commun. Number Theory Phys. (2015), 643–687.[94] A. Goncharov, R. Kenyon. Dimers and cluster integrable systems.
Ann. Sci. c. Norm. Supr. (2013), 747–813.[95] B. Grammaticos, R. G. Halburd, A. Ramani, C.-M. Viallet. How to detect the integrability ofdiscrete systems.
J. Phys. A (2009), 454002, 30 pp.[96] M. Gross and B. Siebert. Mirror symmetry via logarithmic degeneration data, II.
J. AlgebraicGeom. (2010), no. 4, 679–780.[97] M. Gross and B. Siebert. Mirror symmetry via logarithmic degeneration data. I.
J. DifferentialGeom. (2006), no. 2, 169–338.[98] P. Gruber. Only ellipsoids have caustics.
Math. Ann. (1995), 185–194.[99] L. Guglielmi.
Symplectic invariants of integrable Hamiltonian systems with singularities.
PhDThesis, SISSA, 2018.[100] V. Guillemin, S. Sternberg.
The Gel’fand-Cetlin system and quantization of the complex flagmanifolds.
J. Funct. Anal., (1983), 106–128.[101] V. Guillemin, E. Miranda, A. Pires, Symplectic and Poisson geometry on b -manifolds. Adv.Math. (2014), 864–896. 47102] V. Guillemin, E. Miranda, J. Weitsman.
Desingularizing b m -symplectic structures . Int. Math.Res. Not. IMRN, rnx126, 2017, published online, https://doi.org/10.1093/imrn/rnx126.[103] V. Guillemin, E. Miranda, J. Weitsman. On geometric quantization of b -symplectic manifolds . arXiv:1608.08667 , to appear in Adv. Math. .[104] V. Guillemin, E. Miranda, J. Weitsman.
Convexity for Hamiltonian torus actions on b m -symplectic manifolds . to appear in Philos. Trans. Roy. Soc. A .[105] V. Guillemin, E. Miranda, J. Weitsman.
On geometric quantization of b m -symplectic manifolds . arXiv:1801.03762 .[106] V. Guillemin, E. Miranda, A. Pires, G. Scott. Convexity for Hamiltonian torus actions on b -symplectic manifolds . Math. Research Letters (2017), 363–377.[107] V. Guillemin, E. Miranda, A. Pires, G. Scott. Toric actions on b-symplectic manifolds . Int. Math.Res. Not. IMRN 2015, no. 14, 5818–5848.[108] E. Gutkin, S. Tabachnikov.
Billiards in Finsler and Minkowski geometries.
J. Geom. Phys. (2002), 277–301.[109] J. Haantjes. On X m -forming sets of eigenvectors. Nederl. Akad. Wetensch. Proc. Ser. A. (1955) = Indag. Math. (1955), 158–162.[110] R. G. Halburd. Diophantine integrability.
J. Phys. A (2005), L263–L269.[111] M. Hamilton. Locally toric manifolds and singular Bohr-Sommerfeld leaves.
Mem. Amer. Math.Soc. (2010), no. 971, vi+60 pp.[112] M. Hamilton, E. Miranda.
Geometric quantization of integrable systems with hyperbolic singu-larities . Annales Inst. Fourier (2010), 51–85.[113] M. Harada and K. Kaveh. Integrable systems, toric degenerations and Okounkov bodies.
Inven-tiones mathematicae, (2015), 927985[114] G. J. Heckman.
Quantum integrability for the Kovalevsky top.
Indag. Math. (1998), 359–365.[115] H. Hezari. Inverse spectral problems for Schrdinger operators.
Comm. Math. Phys. (2009),1061–1088.[116] S. Hohloch, S. Sabatini, D. Sepe.
From compact semi-toric systems to Hamiltonian S -spaces .Discrete Contin. Dyn. Syst. (2015), 247–281.[117] G. Huang, V. Kaloshin, A. Sorrentino. Nearly circular domains which are integrable close to theboundary are ellipses. arXiv:1705.10601 .[118] A. Iantchenko, J. Sj¨ostrand, M. Zworski.
Birkhoff normal forms in semi-classical inverse prob-lems.
Math. Res. Lett., (2002), 337–362. 48119] A. Ibort, F. Magri, G. Marmo. Bihamiltonian structures and St¨ackel separability.
J. Geom. Phys. (2000), 210–228.[120] N. Innami. Geometry of geodesics for convex billiards and circular billiards.
Nihonkai Math. J., (2002), 73–120.[121] A. Izosimov. Smooth invariants of focus-focus singularities.
Moscow Univ. Math. Bull. (2011),no. 4, 178–180.[122] A. Izosimov. Generalized argument shift method and complete commutative subalgebras in poly-nomial Poisson algebras. arXiv:1406.3777 .[123] A. Izosimov.
Pentagram maps and refactorization in Poisson-Lie groups . arXiv:1803.00726 .[124] V.V. Kalashnikov. Generic integrable Hamiltonian systems on a four-dimensional symplecticmanifold.
Izv. Ross. Akad. Nauk Ser. Mat. (1998), no. 2, 49–74; translation Izv. Math. (1998), 261–285.[125] E. Kalnins, J. Kress, W. Miller. Second-order superintegrable systems in conformally flat spaces.I. Two-dimensional classical structure theory’,II. the classical two-dimensional St¨ackel transform.II. Three-dimensional classical structure theory.
J. Math. Phys. (2005), 053509, 053510, and103507.[126] E. Kalnins, J. Kress, W. Miller. Second order superintegrable systems in conformally flatspaces. IV. The classical 3D St¨ackel transform and 3D classification theory. V. Two- and three-dimensional quantum systems.
J. Math. Phys. (2006), 043514 and 093501.[127] V. Kaloshin, A. Sorrentino. On Local Birkhoff Conjecture for Convex Billiards. arXiv:1612.09194 .[128] V. Kaloshin, K. Zhang.
Density of convex billiards with rational caustics. arXiv:1706.07968 .[129] V. Kaloshin, J. Zhang.
Coexistence of periodic-2 and periodic-3 caustics for nearly circular ana-lytic billiard maps. arXiv:1711.10541 .[130] K. Kaveh.
Toric degenerations and symplectic geometry of projective varieties. arXiv:1508.00316 .[131] B. Khesin, F. Soloviev.
Integrability of higher pentagram maps.
Math. Ann. (2013), 1005–1047.[132] B. Khesin, F. Soloviev.
Non-integrability vs. integrability in pentagram maps.
J. Geom. Phys. (2015), 275–285.[133] B. Khesin, F. Soloviev. The geometry of dented pentagram maps.
J. Eur. Math. Soc. (2016),147–179.[134] A. Kiesenhofer, E. Miranda. Cotangent Models for Integrable Systems.
Comm. Math. Phys. (2017), 1123–1145. 49135] A. Kiesenhofer, E. Miranda, G. Scott.
Action-angle variables and a KAM theorem for b-Poissonmanifolds . J. Math. Pures Appl. (2016), 66–85.[136] A. A. Kirillov.
Geometric quantization , Dynamical systems. IV. Symplectic geometry and its ap-plications, Encycl. Math. Sci. 4, 137-172 (1990); translation from Itogi Nauki Tekh., Ser. Sovrem.Probl. Mat., Fundam. Napravleniya 4, 141–178 (1985).[137] K. Kiyohara.
Two-dimensional geodesic flows having first integrals of higher degree.
Math. Ann. (2001), 487–505.[138] V. N. Kolokoltsov.
Geodesic flows on two-dimensional manifolds with an additional first integralthat is polynomial with respect to velocities.
Math. USSR-Izv. (1983), no. 2, 291–306.[139] M. Kontsevich. Noncommutative identities . arXiv:1109.2469 .[140] Y. Kosmann-Schwarzbach, F. Magri. Poisson-Nijenhuis structures.
Ann. Inst. H. Poincar Phys.Th´eor. (1990), 35–81.[141] B. Kostant. On the Definition of Quantization, G´eom´etrie Symplectique et PhysiqueMath´ematique,
Coll. CNRS, No. 237, Paris, (1975), 187–210.[142] B.S. Kruglikov, V.S. Matveev.
The geodesic flow of a generic metric does not admit nontrivialintegrals polynomial in momenta.
Nonlinearity (2016), 1755–1768.[143] C. Laurent-Gengoux, E. Miranda. Coupling symmetries with Poisson structures . Acta Math.Vietnam. (2013), 21–32.[144] C. Laurent-Gengoux, E. Miranda, P. Vanhaecke. Action-angle coordinates for integrable systemson Poisson manifolds.
Int. Math. Res. Not. IMRN 2011, no. 8, 1839–1869.[145] J. Landsberg.
Exterior differential systems and billiards . Geometry, integrability and quantiza-tion, 35–54, Softex, Sofia, 2006.[146] Y. Le Floch.
Singular Bohr–Sommerfeld conditions for 1D Toeplitz operators: hyperbolic case.
Anal. PDE (2014), 1595–1637.[147] Y. Le Floch. Singular Bohr-Sommerfeld conditions for 1D Toeplitz operators: elliptic case.
Comm. Partial Differential Equations (2014), 213–243.[148] Y. Le Floch, ´A. Pelayo. Spectral asymptotics of semiclassical unitary operators. arXiv:1506.02873.[149] Y. Le Floch, ´A. Pelayo, S. V˜u Ngo. c.
Inverse spectral theory for semiclassical Jaynes-Cummingssystems.
Math. Ann. (2016), 1393–1413.[150] K. Lee.
A step towards the cluster positivity conjecture. arXiv:1103.2726 .[151] L. Lerman, Ya. Umanskii.
The classification of four dimensional Hamiltonian systems and Pois-son actions of R in extended neighborhoods of simple singular points. Math. Sb. (1993),103–138. 50152] N.C. Leung, M. Symington.
Almost toric symplectic four-manifolds.
J. Symp. Geom. (2010),143–187.[153] F. Magri. A simple model of the integrable Hamiltonian equation.
J. Math. Phys. (1978),1156–1162.[154] F. Magri, C. Morosi. A geometrical characterization of integrable Hamiltonian systems through thetheory of Poisson–Nijenhuis manifolds.
Preprint (Universit`a di Milano, Dipartimento di Matem-atica “F. Enriques”, Quaderno S 19/1984). 1984.[155] S.V. Manakov.
Note on the integration of Euler’s equation of the dynamics of an N -dimensionalrigid body. Funct. Anal. Appl. (1976), 328–329.[156] G. Mar´ı Beffa. On integrable generalizations of the pentagram map.
Int. Math. Res. Notices 2015,no. 12, 3669–3693.[157] V. G. Marikhin, V. V. Sokolov.
Pairs of commuting Hamiltonians that are quadratic in momenta.
Teoret. Mat. Fiz. (2006), no. 2, 147–160; translation in Theoret. and Math. Phys. (2006),1425–1436.[158] V. S. Matveev.
Computation of values of the Fomenko invariant for a point of the type “saddle-saddle”of an integrable Hamiltonian system.
Tr. Semin. Vektorn. Tenzorn. Anal, (1993), 75–105[159] V. S. Matveev. Integrable Hamiltonian Systems with two degrees of freedom. Topological structureof saturated neighbourhoods of saddle-saddle and focus points.
Mat. Sb. (1996), no. 4, 29–58.[160] V. S. Matveev, A. A. Oshemkov.
Algorithmic classification of invariant neighborhoods of saddle-saddle points.
Vestnik Moskov. Univ. Ser. I Mat. Mekh. (Moscow University Math. Bull.), ,no. 2, 62 – 65.[161] V. S. Matveev.
Proof of Projective Lichnerowicz-Obata Conjecture.
J. Diff. Geom. (2007),459-502.[162] V. S. Matveev, V. V. Shevchishin. Two-dimensional superintegrable metrics with one linear andone cubic integral . J. Geom. Phys. (2011), 1353–1377.[163] V. S. Matveev. Pseudo-Riemannian metrics on closed surfaces whose geodesics flows admitnontrivial integrals quadratic in momenta, and proof of the projective Obata conjecture for two-dimensional pseudo-Riemannian metrics.
J. Math. Soc. Japan (2012), 107–152.[164] D. McDuff. The moment map for circle actions on symplectic manifolds.
J. Geom. Phys. (1988),149-160.[165] A. Mikhailov, V. Sokolov. Integrable ODEs on associative algebras.
Comm. Math. Phys. (2000), 231–251.[166] E. Miranda.
On symplectic classification of singular Lagrangian foliations.
PhD thesis, Univer-sitat de Barcelona, 2003. 51167] E. Miranda, F. Presas, R. Solha.
Geometric quantization of semitoric systems and almost toricmanifolds. arXiv:1705.06572 .[168] E. Miranda, G. Scott.
The geometry of E -manifolds . arXiv:1802.02959 .[169] E. Miranda, N. T. Zung. Equivariant normal forms for nondegenerate singular orbits of integrableHamiltonian systems . Ann. Sci. Ecole Norm. Sup., (2004), 819–839.[170] A. E. Mironov. Polynomial integrals of a mechanical system on a two-dimensional torus.
Izv.Ross. Akad. Nauk Ser. Mat. (2010), no. 4, 145–156; Englsih translation in Izv. Math. (2010), 805–817.[171] A. E. Mironov. Polynomial integrals of a mechanical system on a two-dimensional torus.
Izv.Ross. Akad. Nauk Ser. Mat. (2010), no. 4, 145–156; translation in Izv. Math. (2010),805–817.[172] A.S. Mischenko, A.T. Fomenko. Euler equations on finite-dimensional Lie groups.
Mathematicsof the USSR-Izvestiya, (1978), no. 2, 371–389.[173] A. Nijenhuis. X n − -forming sets of eigenvectors. Proc. Kon. Ned. Akad. Amsterdam (1951),200–212.[174] T. Nishinou, Y. Nohara, K. Ueda. Toric degenerations of Gelfand-Cetlin systems and potentialfunctions.
Adv. Math., (2010), 648–706.[175] Y. Nohara and K. Ueda.
Toric degenerations of integrable sys- tems on grassmannians andpolygon spaces.
Nagoya Mathematical Journal, (2014), 125–168.[176] P.J. Olver.
Canonical forms and integrability of bi-Hamiltonian systems.
Phys. Lett. A. 1990. , 177–187.[177] P. Olver, V. Sokolov.
Integrable evolution equations on associative algebras.
Comm. Math. Phys. (1998), 245–268.[178] V. Ovsienko, R. Schwartz, S. Tabachnikov.
The pentagram map: a discrete integrable system.
Comm. Math. Phys. (2010), 409–446.[179] V. Ovsienko, R. Schwartz, S. Tabachnikov.
Liouville-Arnold integrability of the pentagram mapon closed polygons.
Duke Math. J. (2013), 2149–2196.[180] ´A. Pelayo.
Hamiltonian and symplectic symmetries: an introduction.
Bull. Amer. Math. Soc., (2017), 383-436[181] ´A. Pelayo. Symplectic spectral geometry of semiclassical operators.
Bull. Belg. Math. Soc. SimonStevin. (2013), 405–415.[182] ´A. Pelayo, T. Ratiu. Circle-valued momentum maps for symplectic periodic flows.
Enseign. Math. (2012), 205–219. 52183] ´A. Pelayo, X. Tang, Vu Ngoc’s Conjecture on focus-focus singular fibers with multiple pinchedpoints , arXiv:1803.00998.[184] ´A. Pelayo, S. Tolman.
Fixed points of symplectic periodic flows.
Ergodic Theory Dynam. Systems. (2011), 1237–1247.[185] ´A. Pelayo, S. V˜u Ngo. c. Semitoric integrable systems on symplectic -manifolds . Invent. Math. (2009), 571–597.[186] ´A. Pelayo, S. V˜u Ngo. c. Constructing integrable systems of semitoric type
Acta Math. (2011),93–125.[187] ´A. Pelayo, S. V˜u Ngo. c.
Symplectic theory of completely integrable Hamiltonian systems.
Bull.Amer. Math. Soc. (2011), 409–455.[188] ´A. Pelayo, S. V˜u Ngo. c. First steps in symplectic and spectral theory of integrable systems.
DiscreteContin. Dyn. Syst. (2012), 3325–3377.[189] ´A. Pelayo, S. V˜u Ngo. c. Semiclassical inverse spectral theory for singularities of focus-focus type.
Comm. Math. Phys. (2014), 809–820.[190] ´A. Pelayo, L. Polterovich, S. V˜u Ngo. c.
Semiclassical quantization and spectral limits of ~ -pseudodifferential and Berezin-Toeplitz operators. Proc. Lond. Math. Soc. (2014), 676–696.[191] H. Poritsky.
The billiard ball problem on a table with a convex boundary – an illustrative dynamicalproblem.
Ann. of Math. (1950), 446–470.[192] F. Pr¨ufer, F. Tricerri, L.Vanhecke. Curvature invariants, differential operators and local homo-geneity.
Trans. Amer. Math. Soc. (1966), 4643–4652.[193] G. Pucacco, K. Rosquist.
Integrable Hamiltonian systems with vector potentials.
J. Math. Phys. (2005), 012701.[194] V. Retakh. Noncommutative cross-ratios.
J. Geom. Phys. (2014), 13–17.[195] S. Rosemann, K. Sch¨obel. Open problems in the theory of finite-dimensional integrable systemsand related fields . J. Geom. Phys. (2015), 396–414.[196] L.G. Rybnikov, The shift of invariants method and the Gaudin model . Funktsional. Anal. iPrilozhen. (2006), no. 3, 30–43; translation in Funct. Anal. Appl. (2006), 188–199.[197] S.T. Sadetov. A proof of the Mishchenko-Fomenko conjecture.
Dokl. Akad. Nauk, (2004),no. 6, 751–754; English translation in Doklady Math., (2004), 634–638.[198] D. Sepe, S. V˜u Ngo. c. Integrable systems, symmetries, and quantization.
Lett. Math. Phys. (2018), 499–571.[199] V. Schastnyy, D. Treschev.
On local integrability in billiard dynamics . Experimental Math., pp.1–7. Published online: 19 Dec 2017, https://doi.org/10.1080/10586458.2017.1409145
The pentagram map.
Experiment. Math. (1992), 71–81.[201] I. Singer. Infinitesimally homogeneous spaces.
Comm. Pure Appl. Math. (1960), 685–697.[202] J. ´Sniatycki. On cohomology groups appearing in geometric quantization , Differential GeometricMethods in Mathematical Physics (1975), pp. 46–66. Lecture Notes in Math., Vol. 570, Springer,Berlin, 1977.[203] F. Soloviev.
Integrability of the pentagram map.
Duke Math. J. (2013), 2815–2853.[204] M. Symington.
Four dimensions from two in symplectic topology . Topology and geometry of man-ifolds, (Athens, GA, 2001), 153–208, Proc. Sympos. Pure Math. 71, Amer. Math. Soc., Providence,RI, 2003.[205] S. Tabachnikov.
Commuting dual billiard maps.
Geom. Dedicata (1994), 57–68.[206] S. Tabachnikov. Billiards.
Panor. Synth. No. 1 (1995), vi+142 pp.[207] S. Tabachnikov.
Introducing projective billiards.
Ergodic Theory Dynam. Systems (1997),957–976.[208] S. Tabachnikov. Remarks on magnetic flows and magnetic billiards, Finsler metrics and a mag-netic analog of Hilbert’s fourth problem . Modern dynamical systems and applications, 233–250,Cambridge Univ. Press, Cambridge, 2004.[209] S. Tabachnikov.
Geometry and billiards.
Student Mathematical Library, 30. Amer. Math.Soc.,Providence, RI, 2005.[210] S. Tabachnikov.
A baker’s dozen of problems.
Arnold Math. J. (2015), 59–67.[211] R.C. Thompson. Pencils of complex and real symmetric and skew matrices.
Linear Algebra Appl. (1991), 323–371.[212] G. Thompson.
Killing tensors in spaces of constant curvature.
J. Math. Phys. (1986), 2693–2699.[213] S. Tolman. Non-Hamiltonian actions with isolated fixed points.
Invent. Math. (2017), 877–910.[214] S. Tolman, J. Weitsman.
On semifree symplectic circle actions with isolated fixed points.
Topology (2000), 299-309.[215] J. Toth. Various quantum mechanical aspects of quadratic forms.
J. Funct. Anal. (1995),1–42.[216] F.J. Turiel.
Classification locale d’un couple de formes symplectiques Poisson-compatibles.
C. R. Acad. Sci. Paris. S´er. I Math. (1989), 575–578.[217] A. Usnich.
Noncommutative cluster mutations.
Dokl. Nats. Akad. Nauk Belarusi (2009), 27–29, 125. 54218] G. Valent. Superintegrable models on Riemannian surfaces of revolution with integrals of anyinteger degree (I).
Regul. Chaotic Dyn. (2017), 319–352.[219] G. Valent. On a class of integrable systems with a cubic first integral.
Comm. Math. Phys. (2010), 631–649.[220] P. Vanhaecke.
Integrable systems in the realm of algebraic geometry . Second edition. LectureNotes in Mathematics, 1638. Springer-Verlag, Berlin, 2001.[221] A. Veselov.
Integrable mappings . Russian Math. Surveys (1991), 1–51.[222] J. Vey. Sur certains systmes dynamiques sparables.
Amer. J. Math. (1978), 591–614.[223] E. B. Vinberg.
Convex homogeneous cones.
Transl. Moscow Math. Soc. (1963), 340–403.[224] K. Vorushilov. Jordan-Kronecker invariants for semidirect sums defined by standard representa-tion of orthogonal or symplectic Lie algebras.
Lobachevskii J. Math. (2017), 1121–1130.[225] S. V˜u Ngo. c On semi-global invariants for focus-focus singularities.
Topology (2003), 365–380.[226] S. V˜u Ngo. c. Quantum monodromy in integrable systems.
Commun. Math. Phys., (1999),465–479.[227] S. V˜u Ngo. c.
Bohr-Sommerfeld conditions for integrable systems with critical manifolds of focus-focus type.
Comm. Pure Appl. Math. (2000), 143–217.[228] S. V˜u Ngo. c. Symplectic techniques for semiclassical completely integrable systems.
In Topologicalmethods in the theory of integrable systems, 241–270. Camb. Sci. Publ., Cambridge, 2006.[229] S. V˜u Ngo. c.
Syst`emes int´egrables semi-classiques: du local au global.
Panoramas et Syhth`eses,No 22, SMF, 2006.[230] A. Weinstein.
The local structure of Poisson manifolds.
J. Diff. Geom. (1983), 523–557.[231] A.Winterhalder. Linear Nijenhuis-Tensors and the construction of integrable systems . arXiv.org:9709008 .[232] T. Wolf, O. Efimovskaya. On integrability of the Kontsevich non-abelian ODE system.
Lett.Math. Phys. (2012), 161–170.[233] N.M.J. Woodhouse.
Geometric quantization, second edition.
Oxford Mathematical Monographs,The Clarendon Press, Oxford University Press, New York, 1992.[234] S. Zelditch.
Inverse spectral problem for analytic domains. II. Z - symmetric domains. Ann. ofMath. (2009), 205–269.[235] P. Zhang.
Algebraic properties of compatible Poisson structures.
Regul. Chaotic Dyn. (2014),267–288.[236] N.T. Zung. Symplectic topology of integrable Hamiltonian systems. I: Arnold-Liouville with sin-gularities.
Compositio Math. (1996), 179–215.55237] N. T. Zung.
A note on degenerate corank-one singularities of integrable Hamiltonian systems.
Comment. Math. Helv. (2000), 271–283.[238] N. T. Zung. Symplectic topology of integrable Hamiltonian systems, II: Topological classification.
Compositio Math.138