Hydrodynamic-type systems describing 2-dimensional polynomially integrable geodesic flows
aa r X i v : . [ m a t h . DG ] D ec Hydrodynamic–type systems describing 2-dimensional polynomiallyintegrable geodesic flows
Gianni Manno ∗ , Maxim V. Pavlov † September 19, 2018
Abstract
Starting from a homogeneous polynomial in momenta of arbitrary order we extract multi-componenthydrodynamic-type systems which describe 2-dimensional geodesic flows admitting the initial polynomial asintegral. All these hydrodynamic-type systems are semi-Hamiltonian, thus implying that they are integrableaccording to the generalized hodograph method. Moreover, they are integrable in a constructive sense aspolynomial first integrals allow to construct generating equations of conservation laws. According to themultiplicity of the roots of the polynomial integral, we separate integrable particular cases.
Keyword:
Integrable geodesic flows, semi-Hamiltonian hydrodynamic systems
MSC 2010:
Introduction
This paper is devoted to the following classical problem: how to extract geodesic flows that are integrableaccording to the Liouville theorem. As usual, the geodesic flow of an n -dimensional (pseudo-)Riemannianmanifold ( M, g ) is locally described by a system of (generally nonlinear) ODEs(1) ¨ x i + Γ ijk ˙ x j ˙ x k = 0 , i = 1 , . . . , n , where x = ( x i ) is a system of coordinates of M , Γ ijk ( x ) are the Christoffel symbols of the Levi-Civita connectionand ˙ x , ¨ x are, respectively, the first and second derivatives of x w.r.t. an external parameter t . We denote by( x i , p i ) the system of coordinates of T ∗ M induced by coordinates ( x i ). It is well known that system (1) can bewritten in Hamiltonian form ˙ x i = ∂H∂p i , ˙ p i = − ∂H∂x i , i = 1 , . . . , n, where(2) H ( x , p ) = 12 g km ( x ) p k p m and p = ( p i ). Since the Hamiltonian function H ( x , p ) does not depend on t explicitly, Hamilton’s equations areLiouville integrable if there exist n − f k ( x , p ) in involution, i.e. { f k , H } = 0 and { f i , f k } = 0,where { f, g } = ∂f∂x i ∂g∂p i − ∂f∂p i ∂g∂x i is the usual Poisson bracket. ∗ Dipartimento di Scienze Matematiche “G. Lagrange”, Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129 Torino, [email protected] † Sector of Mathematical Physics, Lebedev Physical Institute of Russian Academy of Sciences, Leninskij Prospekt 53, 119991Moscow, Russia, Department of Applied Mathematics,National Research Nuclear University MEPHI,Kashirskoe Shosse 31, 115409Moscow, Russia, Department of Mechanics and Mathematics,Novosibirsk State University, 2 Pirogova street, 630090, Novosibirsk,Russia, [email protected]
1n the present paper we shall consider only the case n = 2. In such a case a first integral f ( x , x , p , p )satisfies the equation(3) { f, H } = ∂f∂x ∂H∂p − ∂f∂p ∂H∂x + ∂f∂x ∂H∂p − ∂f∂p ∂H∂x = 0 . A central idea is the research of first integrals f ( x , x , p , p ) that are homogeneous polynomials of degree N in momenta p i , i.e. of the form(4) f = N X m =0 a m ( x , x ) p N − m p m . By substituting the polynomial ansatz (4) into (3) one can derive a quasi-linear system of first order PDEs (seea similar approach, for instance, in [1]).The problem of describing 2-dimensional metrics admitting a polynomial integral is classical: its formulationis due at least to Darboux [2] and it has been solved (both locally and globally) in the case when N is equaleither to 1 or 2. For N = 2 see [3], where metrics admitting quadratic integrals are called of Liouville type:in that paper the author solved the problem posed in [4] of their local characterization (see also [5] for therelationship with superintegrable metrics -in the sense of [6]- and projectively equivalent metrics). For N = 3see [7, 8], where the problem of finding such metrics is solved under particular assumptions.A possible strategy (see, for instance, [9]) is to fix a coordinate system where the metric assumes some specialform, for instance: ds = a ( x , x )( dx ) + ( dx ) (semi-geodesic coordinates), ds = a ( x , x )[( dx ) + ( dx ) ](isothermal coordinates), ds = ( dx ) + a ( x , x ) dx dx + ( dx ) (Chebyshev coordinates), etc. In all thesecases condition (3), with f given by (4), leads to hydrodynamic-type systems of PDEs on the coefficients a m ( x , x ). Integrability of such hydrodynamic-type systems is a separate question.In this paper we present an alternative construction. We suppose that polynomial (4) has N real roots (notnecessarily distinct), so that (4) can be written in the following factorized form(5) f = N Y m =1 ( α m p + α m p ) . By finding a suitable system of coordinates where both (5) and Hamiltonian (2) assume a particular convenientform, by means of condition (3), we arrive to discuss a first-order quasi-linear system of PDEs in the unknownfunctions α m ( x , x ) , α m ( x , x ) and g ik ( x , x ). Our main achievement is a more general ansatz for polynomialintegrals than that presented in [9]: the semi-Hamiltonian hydrodynamic-type system we obtained contains N +1equations, while the hydrodynamic-type system considered in [9] contains just N equations. Furthermore, thesystem we obtained possesses a simple reduction (i.e. the metric is a linear expression in terms of field variables)to the case considered in [9]. In particular, one can select first order quasi-linear systems according to themultiplicity of roots of polynomial (5). Notations and conventions ( M, g ) will be always a 2-dimensional (pseudo-)Riemannian manifold. The symmetric tensor product is denotedby ⊙ . We shall denote by f x the derivative of f w.r.t. x . All results presented in the paper are of local character,meaning that we always work in suitable neighborhoods. For our purposes we need the following proposition, that is well-known.
Proposition 1.1.
If the geodesic flow of ( M, g ) admits a polynomial first integral f of the following form f = ∓ ( α p + α p ) N then it admits also the first linear integral α p + α p . In particular, ( M, g ) admits a Killing vector field. M, g ) admits a first integral f with at least two distinct roots. We shall use the results of nextProposition to get a description of a such geodesic flow in terms of a hydrodynamic-type system. Proposition 1.2.
Let f be as in (5) . Let us suppose that f admits at least two distinct roots, i.e. (6) f = ( α p + α p )( α p + α p ) Q where Q is a homogeneous polynomial of degree N − in momenta, α ( p ) α ( p ) − α ( p ) α ( p ) = 0 , ( α ( p ) , α ( p )) =0 = ( α ( p ) , α ( p )) at a point p ∈ M . Then, in a neighborhood U of the point p , there exists a system ofcoordinates ( x , x ) such that polynomial f , in the induced system of coordinates ( x , x , p , p ) of T ∗ M , assumes the form (7) f = p p N − Y i =1 ( α i p + α i p ) , α kj = α kj ( x , x ) Proof.
In view of the identification p i ≃ ∂ x i , we can write polynomial (6) as follows: f = X ⊙ Y ⊙ Ξwhere X = α ∂ x + α ∂ x , Y = α ∂ x + α ∂ x and Ξ is a ( N − , X and Y , can always be chosen as coordinate lines: in these coordinates theintegral acquires a factor of p p , i.e. polynomial (6) assumes the form (7). Remark 1.3.
The form of any homogeneous polynomial in momenta (in particular polynomial (7) ) does notchange under a transformation (8) x = x ( x ) , x = x ( x ) . Indeed, both ∂ x and ∂ x do not change direction under the above transformation. Proposition 1.4.
If the geodesic flow of ( M, g ) admits a polynomial integral f with the properties described inProposition 1.2, then there exists a system of coordinates ( x i , p i ) where the Hamiltonian (2) has the form (9) H = 12 ǫ p + g p p + 12 ǫ p , g = g ( x , x ) , ǫ i ∈ {− , , } . and the polynomial f the form (7) .Proof. Let f be a polynomial integral of ( M, g ) with two distinct roots. Then, in view of Proposition 1.2, thereexists a system of coordinates where f has the form (7). Equivalently, it is the same of considering form (4)with a = 0 = a N . We shall use this latter notation. If we substitute this f in (3) we obtain a g x = 0 , a N − g x = 0as they are, respectively, the coefficients of p N +11 and p N +12 of left hand side term of (3). If both a and a N − are not zero, then g x = 0 = g x , implying that g = g ( x ) and g = g ( x ). If a = 0, then we obtain a g x = 0 as a g x is the coefficient of p N p of the left hand side of (3), that gives either g x = 0 or a = 0. Ifthe latter case occurs, then we obtain a g x = 0, and so further. A similar reasoning applies, of course, also ifwe start from a N − g x . We conclude that if the initial polynomial integral is not zero, then g x = 0 = g x , sothat(10) g − = g ( x ) ∂ x ⊙ ∂ x + 2 g ( x , x ) ∂ x ⊙ ∂ x + g ( x ) ∂ x ⊙ ∂ x . Now, in view of Remark 1.3, we can use a changing of coordinates (8) to further simplify (10). In particular,we can find a new system of coordinates such that g = ǫ and g = ǫ , with ǫ i ∈ {− , , } . Thus, in thesenew system of coordinates, Hamiltonian H (see (2)) assumes the form (9) (up to renaming function g ).3n our case the metric with upper indices looks precisely like the metric with lower indices in the case ofthe Chebyshev coordinate net, see formula (26) below. Thus, our coordinate system is conformally equivalentto the Chebyshev net. However, our choice of coordinates is very convenient for our further computations. Forinstance we shall show in Section 1.1 that in this coordinate system all further reductions appear in the mostsimple way.To not overload the notation, from now on we shall consider only the Hamiltonian(11) H = 12 p + g p p + 12 p , i.e. (9) with ǫ = ǫ = 1, as the other cases can be treated in the same way. As a possible polynomial integralof the above Hamiltonian, without loss of generality, in view of Proposition 1.4, we can consider(12) f = a p N − p + a p N − p + ... + a N − p p N − + a N − p p N − . Then substituting (12) into (3) we obtain(13) ( p + g p ) f x − p p ( g ) x f p + ( g p + p ) f x − p p ( g ) x f p = 0from which one can derive a quasi-linear system of first order PDEs(14) a ,x + g a ,x = a ( g ) x a k,x + g a k − ,x + g a k,x + a k − ,x = ka k ( g ) x + ( N + 1 − k ) a k − ( g ) x , k = 2 , ..., N − g a N − ,x + a N − ,x = a N − ( g ) x with N unknown functions, i.e. the metric coefficient g and the N − a k of polynomial ansatz(12).A very important property of this system is existence of simple reductions based on the multiplicity of theroots of polynomial (12). For instance: a k = 0 , k = 1 , , ..., K < N − k = N − , N − , ..., K < N − N = 5, we have the following full list of distinguish reductions: a = 0; a = a = 0; a = a = a = 0; a = a = 0; a = a = a = 0. Thus, instead to investigate each particular case, one canconcentrate on the generic system (14) only.However this system is written in a non-evolutionary form. By this reason, in Section 2, we rewrite it in anevolutionary form by an appropriate reciprocal transformation. Here we consider system (14) in the case of polynomial integrals of third and fourth degree. We also brieflydiscuss its reductions according to the multiplicity of roots of the polynomial integral.
Let f be a homogeneous polynomial of a third degree in momenta. We already seen that, if f is a perfect cube,the metric admitting such f as an integral it admits a Killing vector field (see Proposition 1.1). So, let us thenconsider the case when polynomial (5) has at least two distinct roots. In this case, in view of Proposition 1.2,a normal form of such polynomial is (12) with N = 3, i.e.(15) a p p + a p p , a , a ∈ C ∞ ( M ) . that, in view of system (14), is an integral of Hamiltonian (11) iff(16) a ,x + a ,x g = ( g ) x a a ,x g + a ,x + a ,x + a ,x g = 2( g ) x a + 2( g ) x a a ,x g + a ,x = ( g ) x a
4f polynomial f admits two coincident roots, then its normal form is (15) with a = 0 (or, equivalently, a = 0)and, correspondingly, the quasi-linear system to be considered is (16) with either a = 0 or a = 0. Let f be a homogeneous polynomial of fourth degree in momenta. Let us suppose that f admits at least twodistinct roots, otherwise ( M, g ) admits a Killing vector field (see again Proposition 1.1). In this case, in viewof Proposition 1.2, a normal form of such polynomial is (12) with N = 4, i.e.(17) a p p + a p p + a p p , a , a , a ∈ C ∞ ( M ) . that, in view of system (14), is an integral of Hamiltonian (11) iff(18) a ,x + a ,x g = a ( g ) x a ,x + a ,x + a ,x g + a ,x g = 2 a ( g ) x + 3 a ( g ) x a ,x + a ,x + a ,x g + a ,x g = 3 a ( g ) x + 2 a ( g ) x a ,x + a ,x g = a ( g ) x According to the multiplicity of roots of polynomial (17), we have several possible reductions of system (18).In fact, by arguing as in the end of Section 1.1.1, if in system (18) we put a = a = 0, then we obtain thesystem describing metrics admitting a polynomial integral with three coincident roots. If we put a = a = 0,we obtain the system describing metrics admitting a polynomial integral with two coincident roots. Finally, ifwe put a = 0, we obtain the system describing metrics admitting a polynomial integral with two coincidentroots (and two distinct). If no particular assumption is imposed on a , a , a , then, generically, the above systemdescribes metrics admitting polynomial integrals with all distinct roots. In this section we are going to investigate integrability of hydrodynamic-type system (14) selected by theHamiltonian (11) and the polynomial ansatz (12). For our further research we need first to reduce this systemto an evolutionary form. This is possible by finding an appropriate reciprocal transformation; below we presenta constructive algorithm: • We derive the Liouville equation from commutativity of the Hamiltonian and the first integral (3); • we introduce an appropriate reciprocal transformation (to semi–geodesic coordinates); • under this transformation we recompute the metric, the Hamilton-Jacobi equation, the Liouville equation,momenta and finally the hydrodynamic–type system (14); • we discuss the existence of infinite set of conservation laws; • we prove the diagonalizability of this hydrodynamic–type system; • as an example, we shall consider the two component case.Polynomial ansatz (12) can be written in the form (see more details in [10, 11])(19) f = (cid:0) p + 2 g p p + p (cid:1) N/ λ ( s, x , x ) , where s = p /p and the function λ ( s, x , x ) satisfies (13):(20) (1 + g s ) λ x + ( g + s ) λ x + (cid:0) s ( g ) x − s ( g ) x (cid:1) λ s = 0 . p = (1 + 2 g s + s ) − / (instead of the variable s ), equation (20) reduces to the canonical form (22) λ x = { λ, H } = H p λ x − λ p H x , where now the function λ depends on p via formula (21) and the Hamiltonian function is(23) H = p (( g ) − p + 1 − g p. Equation (22) plays an important role in the theory of integrable hydrodynamic chains and semi-Hamiltonianhydrodynamic-type systems (see, for instance, [13, 14]). The existence of representation (22) for hydrodynamic-type system (14) means that such a system is integrable (or semi-Hamiltonian), i.e. it admits infinitely many con-servation laws, commuting flows and particular solutions (see more details in [15, 16]). The integrability proce-dure means that instead of the function λ ( x , x , p ) we consider the function λ ( p, g , a ( x , x ) , . . . , a N − ( x , x ))whose existence is equivalent to the integrability of some overdetermined system (now known as the Gibbons-Tsarev system [17]). Substitution of one of its particular solutions (see (12)) λ ( s, x , x ) = (cid:0) p + 2 g p p + p (cid:1) − N/ f ( x , x , p , p )= ( p + 2 g p p + p ) − N/ ( a p N − p + a p N − p + ... + a N − p p N − + a N − p p N − )= (1 + 2 g s + s ) − N/ ( a s + a s + ... + a N − s N − + a N − s N − )into (22) creates the hydrodynamic–type system (14). The Liouville–type equation (22) takes the form aHamilton–Jacobi equation(24) p x = H x where differentiation of H (given by (23)) w.r.t. x means to differentiate not only g but also p , that now is adependent function on ( x , x , λ ). So, now the function p is a generating function of conservation law densitiesfor hydrodynamic–type system (14), while the Hamilton-Jacobi equation (24) plays the role of the generatingequation of the corresponding conservation laws (w.r.t. the parameter λ ). In this section we are going to consider the main system of our interest (14) in another coordinate systemthat is more convenient for our further computations. Instead of our original coordinates (conformally equiva-lent Chebyshev coordinates, see the discussion before formula (11)), we introduce the so called semi–geodesic coordinates ( x, y ) (see more details in [1]), which we determine by virtue of the reciprocal transformation(25) dy = 1 a N − dx − g a N − dx , dx = dx , where the potential function y follows from the third equation of system (14) written in the conservative form (cid:18) a N − (cid:19) x + (cid:18) g a N − (cid:19) x = 0 , i.e. y x = 1 a N − , y x = − g a N − . Classifications of such Hamiltonian equations is given in [12]. ds = 11 − ( g ) (cid:0) ( dx ) − g dx dx + ( dx ) (cid:1) , in these semi–geodesic coordinates ( x, y ), assumes the form(27) ds = ( dx ) + a N − − ( g ) ( dy ) . Correspondingly, the Hamilton–Jacobi equation (see (24))(28) p x = (cid:16)p (( g ) − p + 1 − g p (cid:17) x becomes (see again [1] and other details in [11])(29) ˜ p y = a N − p − ( g ) p − ˜ p ! x , where we define ˜ p as follows(30) ˜ p = p (( g ) − p + 1(we remind that the variable p has been defined by (21)). In fact, the conservation law (28) can be written inthe potential form dξ = pdx + (cid:16)p (( g ) − p + 1 − g p (cid:17) dx . Under the inverse reciprocal transformation (see (25))(31) dx = g dx + a N − dy , dx = dx we obtain dξ = pa N − dy + p (( g ) − p + 1 dx . The compatibility condition ( ξ x ) y = ( ξ y ) x leads to (29) where ˜ p is given by (30). To recompute first integral(12) via new coordinates ( x, y ) we need first to recompute the corresponding momenta. To do this we use theidentity p dx + p dx = ˜ p dx + ˜ p dy . In this case(32) ˜ p = p + g p , ˜ p = a N − p , then the first integral (12) takes the form(33) f = ˜ p (˜ p ) N − + ˜ a (˜ p ) (˜ p ) N − + ˜ a (˜ p ) (˜ p ) N − + ... + ˜ a N − (˜ p ) N − ˜ p + ˜ a N − (˜ p ) N . Note that all coefficients ˜ a k are linear functions with respect to a m and polynomial functions with respectto g and ( a N − ) − . Taking into account that first integral (33) can be written in the factorized form f = (˜ p ) N N − Y m =1 (cid:18) ˜ p ˜ p − ˜ b m (cid:19) , we introduce (cf. (19)) the function˜ λ (˜ s, x, y ) = s + 1 − ( g ) a N − ! − N/ N − Y m =1 (cid:18) s − ˜ b m (cid:19) , s = ˜ p / ˜ p . For our further convenience we define an appropriate independent variable q instead of ˜ s :(34) q = a N − p − ( g ) s . Then we obtain the equation of the Riemann surface(35) ˜ λ ( x, y, q ) = a − / (1 + q ) − N/ N − Y m =1 ( q − b m ) , where ˜ b k = a / b k and a − / = a N − p − ( g ) . Under the reciprocal transformation (25) Liouville equation (22) takes the form(36) ˜ λ y = a − / q ˜ λ x + (1 + q )˜ λ q ( a − / ) x where (see (32) and (34))(37) ˜ p = q q , q = ˜ p − ˜ p . The Liouville–type equation (36) takes again the form of Hamilton–Jacobi equation (29) (see (24)), that wasalready investigated in [11]. Substitution (35) into (36) yields the hydrodynamic-type system (38) a y = 2 a / (cid:18) N − P m =1 b m (cid:19) x + a − / a x (cid:18) N − P m =1 b m (cid:19) ( b k ) y = a − / b k ( b k ) x − [1 + ( b k ) ]( a − / ) x , k = 1 , , ..., N − . As we mentioned above, this system is integrable by the Generalized Hodograph Method (see [15, 16]). Thismeans that any solution of this system determines an integrable geodesic flow (see details in [11]). We emphasizethat description of integrable geodesic flows selected by Hamiltonian (11) and by homogeneous polynomial firstintegral (12) reduces to the integrability of the hydrodynamic–type system (38). Moreover, once solutions ofthis system are found, one can choose any conservation law density (see (37), here instead of p we use h k andinstead of q we use b k ) h k = b k p b k ) to determine elements of the conservation law (for any index k , the density and the flux correspondingly) g = h k , a N − = r − ( h k ) a of the inverse reciprocal transformation (31). Since this transformation can be determined for any index k , wehave N − x , x ). Here we briefly discuss the integrabilityof system (38). Its integrability is based on two important properties: existence of Riemann invariants (in thesecoordinates any hydrodynamic–type system takes a diagonal form) and existence of infinitely many conservationlaws. First we explain how to compute these conservation laws. By introducing the so called moments B k = 1 k + 1 N X m =1 ( b m ) k +1 , another hydrodynamic-type system was found in [1]. The approach presented in this Section was established in [14] and utilizedin [10] for the hydrodynamic-type system derived by V.V. Kozlov in [18], where that hydrodynamic-type system describes classicalmechanical systems with one-and-a-half degree of freedom and with polynomial first integrals. a y = 2 a / B x + a − / B a x B y = a − / B x − ( N + 2 B )( a − / ) x B ky = a − / B k +1 x + ( kB k − + ( k + 2) B k +1 )( a − / ) x , k = 1 , , . . . System (38) has infinitely many polynomial conservation laws whose densities and fluxes depend only on a andmoments B k . One can derive them either iteratively step by step or from the equation of the Riemann surface(35) (see details below and [10, 11]). For instance, first two conservation laws are a y = (cid:16) a / B (cid:17) x , (cid:16) B a / (cid:17) y = (cid:18) a (cid:18)
32 ( B ) + B + N (cid:19)(cid:19) x . Moreover, system (38) is diagonalizable, i.e. it can be put in the form(39) r it = µ i ( r ) r ix i = 1 , . . . , N where r k are Riemann invariants, determined by the condition ˜ λ q = 0, i.e., they are branch points of the Riemannsurface determined by the equation (35). More precisely, r k ( b ) = ˜ λ ( b , q ) | q = q k ( b ) , where b = ( b , . . . , b N − ), i.e.,in our case, the N distinct roots q k ( b ) determined by the condition N q q = N − X m =1 q − b m and then substituted into the equation of Riemann surface (35) give the Riemann invariants r k ( x, y ) = a − / (1 + q k ) − N/ N − Y m =1 ( q k − b m ) . So, if q → q k ( b ), ˜ λ ( b , q ) → r k ( b ), ˜ λ q →
0, then (36) leads to (39), with characteristic velocities µ k ( r ( a, b )) = a − / q k ( b ). Hydrodynamic–type systems which are simultaneously diagonalizable and possess infinitely manyconservation laws are integrable by the Generalized Hodograph Method (see [15, 16]). For instance, if N = 2,then characteristic velocities are µ = a − / (cid:16) b + p ( b ) + 1 (cid:17) , µ = a − / (cid:16) b − p ( b ) + 1 (cid:17) ;and the corresponding Riemann invariant are r = 12 a − / b + p ( b ) + 1 , r = 12 a − / b − p ( b ) + 1 . Thus, hydrodynamic-type system (39) becomes r t = − r r x , r t = − r r x . This system (as well as the corresponding metric written in semi-geodesic coordinates) is discussed in detail in[11].
Remark 2.1.
The first integral investigated in [1] and [11] is different (cf. (33) ): f = (˜ p ) N + ˜ p (˜ p ) N − + ˜ a (˜ p ) (˜ p ) N − + ˜ a (˜ p ) (˜ p ) N − + ... + ˜ a N − (˜ p ) N − ˜ p + ˜ a N − (˜ p ) N . However the integration procedure based on the Generalized Hodograph Method (see more details in [15, 16]) isprecisely the same as in [11]. Conclusion
In this paper we presented an approach, described in details in Section 1, that allows to reduce the problem of thedescription of integrable geodesic flows selected by homogeneous polynomial first integrals to the integrabilityof semi–Hamiltonian hydrodynamic–type systems possessing a variety of inequivalent reductions. In fact, thepolynomial ansatz (12) for the first integral depends on two natural numbers: one of them is the degree of thepolynomial and the other is the number of its non-zero coefficients. Moreover, the considered hydrodynamic–type systems can be written in evolutionary form by an appropriate reciprocal transformation to semi-geodesiccoordinates. All of them can be integrated by the Generalized Hodograph Method. However, even in the twocomponents cases, such solutions can be presented just in implicit form. Nevertheless, this two componentcase is very interesting since characteristic velocities can be expressed explicitly via Riemann invariants. Thisparticular research will be the topic of a future investigation.
Acknowledgements
The authors thank Andrey Mironov and Sergey Tsarev for important discussions. Both authors were partiallysupported by the grant “Finanziamento giovani studiosi - Metriche proiettivamente equivalenti, equazioni diMonge–Amp`ere e sistemi integrabili”, University of Padova 2013-2015 and by the project “FIR (Futuro inRicerca) 2013 - Geometria delle equazioni differenziali”. The second author was partially supported by thegrant of Presidium of RAS “Fundamental Problems of Nonlinear Dynamics” and by the RFBR grant 15-01-01671-a. The first author is member of G.N.S.A.G.A. of I.N.d.A.M.
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