Polynomials with Multiple Zeros and Solvable Dynamical Systems including Models in the Plane with Polynomial Interactions
aa r X i v : . [ m a t h - ph ] M a r Polynomials with Multiple Zerosand Solvable Dynamical Systemsincluding Models in the Planewith Polynomial Interactions
Francesco Calogero a,b, and Farrin Payandeh a,c, a Physics Department, University of Rome ”La Sapienza”, Rome, Italy b INFN, Sezione di Roma 1 c Department of Physics, Payame Noor University (PNU), PO BOX19395-3697 Tehran, Iran [email protected], [email protected] f [email protected], [email protected] Abstract
The interplay among the time-evolution of the coefficients y m ( t ) and the zeros x n ( t ) of a generic time-dependent (monic) polynomial provides a conve-nient tool to identify certain classes of solvable dynamical systems. Recentlythis tool has been extended to the case of nongeneric polynomials characterizedby the presence, for all time, of a single double zero; and subsequently signifi-cant progress has been made to extend this finding to the case of polynomialsfeaturing a single zero of arbitrary multiplicity. In this paper we introduce anapproach suitable to deal with the most general case, i. e. that of a nongeneric time-dependent polynomial with an arbitrary number of zeros each of whichfeatures, for all time, an arbitrary (time-independent) multiplicity. We thenfocus on the special case of a polynomial of degree 4 featuring only 2 differentzeros and, by using a recently introduced additional twist of this approach, wethereby identify many new classes of solvable dynamical systems of the followingtype: ˙ x n = P ( n ) ( x , x ) , n = 1 , , with P ( n ) ( x , x ) two polynomials in the two variables x ( t ) and x ( t ). Notation 1.1 . Hereafter t generally denotes time (the real independent vari-able); (partial) derivatives with respect to time are denoted by a superimposeddot, or in some case by appending as a subscript the independent variable t preceded by a comma; all dependent variables such as x , y , z (often equipped1ith subscripts) are generally assumed to be complex numbers, unless otherwiseindicated (it shall generally be clear from the context which of these and otherquantities depend on the time t , as occasionally—but not always— explicitly indicated); parameters such as a , b, c, α, β, γ, A , etc. (often equipped withsubscripts) are generally time-independent complex numbers; and indices suchas n , m , j , ℓ are generally positive integers , with ranges explicitly indicated orclear from the context. (cid:4) Some time ago the idea has been exploited to identify dynamical systemswhich can be solved by using as a tool the relations between the time evolutionsof the coefficients and the zeros of a generic time-dependent polynomial [1]. Thebasic idea of this approach is to relate the time-evolution of the N zeros x n ( t )of a generic time-dependent polynomial p N ( z ; t ) of degree N in its argument z,p N ( z ; t ) = z N + N X m =1 (cid:2) y m ( t ) z N − m (cid:3) = N Y n =1 [ z − x n ( t )] , (1)to the time-evolution of its N coefficients y m ( t ). Indeed, if the time evolutionof the N coefficients y m ( t ) is determined by a system of Ordinary DifferentialEquations (ODEs) which is itself solvable , then the corresponding time-evolutionof the N zeros x n ( t ) is also solvable , via the following 3 steps: (i) given the initialvalues x n (0) , the corresponding initial values y m (0) can be obtained from the explicit formulas expressing the coefficients y m of a polynomial in terms of its zeros x n reading (for all time, hence in particular at t = 0) y m ( t ) = ( − m N X ≤ n ODEs, postponing the treatmentof dynamical systems characterized by higher-order ODEs (see Section 6 ). (cid:4) A new twist of this approach was then provided by its extension to non-generic polynomials featuring—for all time— multiple zeros. The first step inthis direction focussed on time-dependent polynomials featuring for all time a single double zero [10]; and subsequently significant progress has been madeto treat the case of polynomials featuring a single zero of arbitrary multiplic-ity [11]. In Section 2 of the present paper a convenient method is providedwhich is suitable to treat the most general case of polynomials featuring an ar-bitrary number of zeros each of which features an arbitrary multiplicity . Whileall these developments might appear to mimic scholastic exercises analogous tothe discussion among medieval scholars of how many angels might dance simul-taneously on the point of a needle, they do indeed provide new tools to identify new dynamical systems featuring interesting time evolutions (including systemsdisplaying remarkable behaviors such as isochrony or asymptotic isochrony : seefor instance [10] [11]); dynamical systems which—besides their intrinsic mathe-matical interest—are quite likely to play significant roles in applicative contexts.Such developments shall be reported in future publications. In the presentpaper we focus on another twist of this approach to identify new solvable dy-namical systems which was introduced quite recently [12]. It is again based onthe relations among the time-evolution of the coefficients and the zeros of time-dependent polynomials [6] [7] with multiple roots (see [10], [11] and above); but(as in [12]) by restricting attention to such polynomials featuring only zeros .3gain, this might seem such a strong limitation to justify the doubt that theresults thereby obtained be of much interest. But the effect of this restrictionis to open the possibility to identify algebraically solvable dynamical modelscharacterized by the following system of 2 ODEs,˙ x n = P ( n ) ( x , x ) , n = 1 , , (4)with P ( n ) ( x , x ) two polynomials in the two dependent variables x ( t ) and x ( t ); hence systems of considerable interest, both from a theoretical and anapplicative point of view (see [12] and references quoted there). This develop-ment is detailed in the following Section 3 by treating a specific example. In Section 4 we report—without detailing their derivation, which is rather obviouson the basis of the treatment provided in Section 3 —many other such solvable models (see (4); but in some cases the right-hand side of these equations arenot quite polynomial); and a simple technique allowing additional extensions ofthese models—making them potentially more useful in applicative contexts—isoutlined in Section 5 , by detailing its applicability in a particularly interestingcase. Hence researchers primarily interested in applications of such systems ofODEs might wish to take first of all a quick look at these 2 sections.Finally, Section 6 outlines future developments of this research line; andsome material useful for the treatment provided in the body of this paper isreported in 2 Appendices. N zeros, each of arbitrary multiplicity In this Section N different zeros x n ( t ), each of which with the arbitrarily assigned (ofcourse time-independent) multiplicity µ n . They are of course defined as follows: P M ( z ; t ) = z M + M X m =1 (cid:2) y m ( t ) z M − m (cid:3) = N Y n =1 { [ z − x n ( t )] µ n } . (5a)Here the N positive integers µ n are a priori arbitrary. It is obvious that thisformula implies that the degree of this polynomial P M ( z ; t ) is M = N X n =1 ( µ n ) . (5b)It is plain that there exist explicit formulas—generalizing (2)—expressingthe M coefficients y m ( t ) in terms of the N zeros x n ( t ); for instance clearly y ( t ) = − N X n =1 [ µ n x n ( t )] , y M ( t ) = ( − M N Y n =1 { [ x n ( t )] µ n } , (6)4nd see other examples below.It is also plain that, while N zeros x n and their multiplicities µ n can be arbitrarily assigned in order to define the polynomial (5), this is not the case forthe M coefficients y m : generally—for any given assignment of the N multiplic-ities µ n —only N of them can be arbitrarily assigned, thereby determining (viaalgebraic operations) the N zeros x n and the remaining M − N other coefficients y m . Remark 2-1 . The generic polynomial (1), of degree N and featuring N different zeros x n and N coefficients y m , generally implies that the set of its N coefficients y m is an N -vector ~y = ( y , ..., y N ) , while the set of its N zeros x n isinstead an unordered set of N numbers x n . This however is not quite true in thecase of a time-dependent generic polynomial which features—as those generallyconsidered in this paper—a continuous time-dependence of its coefficients and zeros ; then the set of its N zeros x n (0) at the initial time t = 0 should begenerally considered an unordered set, but for all subsequent time, t > 0, theset of its N zeros x n ( t ) is an ordered set, the assignment of the index n to x n ( t ) being no more arbitrary but rather determined by continuity in t (at leastprovided during the time evolution no collision of two or more different zeros occur, in which case the identities of these zeros get to some extent lost becausetheir identities may be exchanged, becoming undetermined ).The situation is quite different in the case of a nongeneric polynomial suchas (5): then zeros having different multiplicities are intrinsically different, forinstance if all the multiplicities µ n are different among themselves, µ n = µ ℓ if n = ℓ , then clearly the set of the N zeros x n is an ordered set (hence an N -vector).We trust the reader to understand these rather obvious facts and thereforehereafter we refrain from any additional discussion of these issues. (cid:4) Our task now is to identify—for the special class of nongeneric polynomials(5)— equivalent relations to the identities (3), to be then used in order toidentify new solvable dynamical systems.The first step is to time-differentiate once the formula (5), getting the rela-tions P M,t ( z ; t ) = M X m =1 (cid:2) ˙ y m ( t ) z M − m (cid:3) = − N X n =1 µ n ˙ x n ( t ) [ z − x n ( t )] µ n − N Y ℓ =1 , ℓ = n { [ z − x ℓ ( t )] µ ℓ } . (7) Remark 2-2 . Hereafter, in order to avoid clattering our presentation withunessential details, we occasionally make the convenient assumption that allthe numbers µ n be different among themselves; the diligent reader shall haveno difficulty to understand how the treatment can be extended to include casesin which this simplifying assumption does not hold—indeed in the specific ex-amples discussed below we will include in our treatment also cases in whichthis simplification is not valid, taking appropriate care of such cases. And we5lso assume—without loss of generality—that the numbers µ n are ordered indecreasing order, µ n ≥ µ n +1 . (cid:4) Our next step is to z -differentiate µ times the above formulas, firstly with µ = 0 , , , ..., µ n − µ = µ n − 1; and then set z = x n (for eachvalue of n = 1 , , ..., N ). There clearly thereby obtain the following formulas: M − µ X m =1 (cid:26) ˙ y m ( t ) (cid:20) ( M − m )!( M − m − µ )! (cid:21) [ x n ( t )] M − m − µ (cid:27) = 0 ,µ = 0 , , ..., µ n − , n = 1 , , ..., N ; (8a) M − µ n +1 X m =1 (cid:26) ˙ y m ( t ) (cid:20) ( M − m )!( M − m − µ n + 1)! (cid:21) [ x n ( t )] M − m − µ n +1 (cid:27) = − ( µ n !) ˙ x n ( t ) N Y ℓ =1 , ℓ = n { [ x n ( t ) − x ℓ ( t )] µ ℓ } ,n = 1 , , ..., N . (8b)The second set, (8b), yields the following N expressions of the time-derivativesof the N zeros x n ( t ) in terms of the time-derivatives of the M coefficients y m ( t ):˙ x n ( t ) = − µ n ! N Y ℓ =1 , ℓ = n [ x n ( t ) − x ℓ ( t )] µ ℓ − ·· M − µ n +1 X m =1 (cid:26) ˙ y m ( t ) (cid:20) ( M − m )!( M − m − µ n + 1)! (cid:21) [ x n ( t )] M − m − µ n +1 (cid:27) ,n = 1 , , ..., N . (9)The first set, (8a), consists of linear relations among the M time-derivatives˙ y m ( t ) of the M coefficients y m ( t ): and it is easily seen, via (5b), that there arealtogether N X n =1 ( µ m − 1) = M − N (10)such relations. So one can select N quantities ˙ y m ( t )—let us hereafter callthem ˙ y ˜ m ( t )—and compute, from the M − N linear equations (8a), all the other M − N quantities ˙ y m ( t ) with m = ˜ m as linear expressions in terms of theseselected N quantities ˙ y ˜ m ( t ). The goal of expressing the N time-derivatives ˙ x n as linear equations—somehow analogous to the identities (3)—in terms of the N time-derivatives ˙ y ˜ m ( t ) of N , arbitrarily selected, coefficients y ˜ m ( t ) is therebyfinally achieved. Indeed the task of expressing the M − N quantities ˙ y m ( t ) with m = ˜ m in terms of the N quantities ˙ y ˜ m ( t )—and of course the N zeros x n ( t )—can in principle be implemented explicitly as it amounts to solving the M − N inear equations (8) for the M − N unknowns ˙ y m ( t ), with the N quantities˙ y ˜ m ( t ) playing there the role of known quantities; clearly implying that theresulting expressions of the M − N quantities ˙ y m ( t ) are linear functions of the N quantities ˙ y ˜ m ( t ). And the insertion of these linear expressions of the M − N quantities ˙ y m ( t ) (with m = ˜ m ) in terms of the N quantities ˙ y ˜ m ( t ) in the N formulas (9) fulfils our goal.The actual implementation of this development must of course be performedon a case-by-case basis, see below. In the special case with only one multiplezero—and if moreover the indices ˜ m are assigned their first N values, i. e.˜ m = 1 , , ..., N —these results shall reproduce the results of the path-breakingpaper [11], which was confined to the treatment of this special case.The outcomes of these developments are detailed, in the special case with N = 2 and M = 4, in the following Section 3 ; for the motivation of this drasticrestriction see below. N = 2 , M = 4 case In the special case with N = 2 the formula (9) simplifies, reading (see (5b))˙ x = − [ µ ! ( x − x ) µ ] − µ X m =1 (cid:26) ˙ y m (cid:20) ( µ + µ − m )!( µ − m + 1)! (cid:21) ( x ) µ − m +1 (cid:27) , (11a)˙ x = − [ µ ! ( x − x ) µ ] − µ X m =1 (cid:26) ˙ y m (cid:20) ( µ + µ − m )!( µ − m + 1)! (cid:21) ( x ) µ − m +1 (cid:27) . (11b)Let us moreover restrict attention to the case with M = 4 , as the case with M = 3 (implying µ = 2 , µ = 1: see Remark 2-2 ) has been already discussedin [10] and [12] and the case with M = 4 is sufficiently rich (see below) todeserve a full paper.In the case with M = 4 there are 2 possible assignments of the 2 parameters µ n : (i) µ = 3 , µ = 1; (ii) µ = µ = 2 (see Remark 2-2 , and note that wenow include also the case with µ = µ ). µ = 3 , µ = 1 In this case (5a) clearly implies the following expressions of the 4 coefficient s y m ( t ) in terms of the 2 zeros x n ( t ): y = − (3 x + x ) , y = 3 x ( x + x ) ,y = − ( x ) ( x + 3 x ) , y = ( x ) x . (12) Remark 3.1-1 . Note that these formulas imply that x and x can becomputed (in fact, explicitly !) from y m and y m —with m = 1 , , m 4) by solving an algebraic equation of degree m . (cid:4) y ( x ) + ˙ y ( x ) + ˙ y x + ˙ y = 0 , (13a)3 ˙ y ( x ) + 2 ˙ y x + ˙ y = 0 (13b)(note that in this case only the formulas with n = 1 are present).And the formulas (11) read˙ x = − x ˙ y + ˙ y x − x ) , ˙ x = ( x ) ˙ y + ( x ) ˙ y + x ˙ y + ˙ y ( x − x ) . (14)There are now 6 possible different assignments for the indices ˜ m :˜ m = 1 , 2; ˜ m = 1 , 3; ˜ m = 1 , 4; ˜ m = 2 , 3; ˜ m = 2 , 4; ˜ m = 3 , , (15a)to which there correspond the 6 complementary assignments m = 3 , m = 2 , m = 2 , m = 1 , m = 1 , m = 1 , . (15b)Let us list below the 6 corresponding versions of the ODEs (14):˙ x = − x ˙ y + ˙ y x − x ) , ˙ x = (2 x + x ) ˙ y + ˙ y ( x − x ) , (16a)˙ x = − x ) ˙ y − ˙ y x ( x − x ) , ˙ x = x ( x + 2 x ) ˙ y − ˙ y x ( x − x ) , (16b)˙ x = − ( x ) ˙ y + ˙ y x ) ( x − x ) , ˙ x = ( x ) x ˙ y + ˙ y ( x ) ( x − x ) , (16c)˙ x = x ˙ y + ˙ y x ( x − x ) , ˙ x = − x ( x + 2 x ) ˙ y + (2 x + x ) ˙ y x ) ( x − x ) , (16d)˙ x = ( x ) ˙ y − y x ) ( x − x ) , ˙ x = − ( x ) x ˙ y − (2 x + x ) ˙ y x ) ( x − x ) , (16e)˙ x = − x ˙ y + 3 ˙ y x ) ( x − x ) , ˙ x = x x ˙ y + ( x + 2 x ) ˙ y ( x ) ( x − x ) . (16f)Next, let us focus to begin with—in order to explain our approach—on thefirst, (16a), of the 6 formulas (16). Assume moreover that the 2 quantities y ( t )and y ( t ) evolve according to the following solvable system of ODEs:˙ y = f ( y , y ) , ˙ y = f ( y , y ) . (17)It is then clear—via the identities (12)—that we can conclude that the dynamicalsystem ˙ x = − [3 ( x − x )] − [3 x f ( − (3 x + x ) , x ( x + x ))+ f ( − (3 x + x ) , x ( x + x ))] , (18a)8 x = ( x − x ) − [(2 x + x ) f ( − (3 x + x ) , x ( x + x ))+ f ( − (3 x + x ) , x ( x + x ))] , (18b)is as well solvable .While this is in itself an interesting result—to become more significant forexplicit assignments of the 2 functions f ( y , y ) and f ( y , y ) (see below)—anadditional interesting development emerges if—following the approach of [12]—we now assume the 2 functions f ( y , y ) and f ( y , y ) to be both polynomial in their 2 arguments and moreover such that3 xf (cid:0) − x, x (cid:1) + f (cid:0) − x, x (cid:1) = 0 ; (19)a restriction that is clearly sufficient to guarantee that the right-hand sides ofthe equations of motion (21) become polynomials in the 2 dependent variables x ( t ) and x ( t ) (since the numerators in the right-hand sides of the 2 ODEs(21) are then both polynomials in the variables x and x which vanish when x = x = x and which therefore contain the factor x − x ).An representative example of such functions is f ( y , y ) = α + α y , f ( y , y ) = β y + β ( y ) , (20a)with (see (19)) α = 4 β , α = 32 β solvable, see—up to trivial rescalings of some parameters—the solution in terms of Jaco-bian elliptic functions in Example 1 in [12], and, below, in Subsection CaseA.3.1 of Appendix A .The conclusion is then that the dynamical system˙ x = a + b h x ) + 10 x x + ( x ) i , (21a)˙ x = a + b h 17 ( x ) + 2 x x − x ) i , (21b)with a = − β / b = − β / arbitrary parameters, is solvable : indeedthe solution of its initial-values problem—to evaluate x ( t ) and x ( t ) from arbitrarily assigned initial values x (0) and x (0)—are ( explicitly !) yielded bythe solution of a quadratic algebraic equation the coefficients of which involvethe Jacobian elliptic function µ sn( λt + ρ, k ) with the 4 parameters µ, λ, ρ, k given by simple formulas in terms of the 2 initial data x (0) and x (0) andthe 2 a priori arbitrary parameters a and b . (The interested reader can easilyobtain all the relevant formulas from the treatment given above, comparing itif need be with the analogous treatment provided in Example 1 of [12]; or seebelow Subsection 4.7) . Remark 3.1-2 . An equivalent—indeed more direct—way to identify the solvable dynamical system (21) as corresponding to the solvable dynamical sys-tem ˙ y = 43 β − β y , ˙ y = β y + β ( y ) (22)9see (17) and (20)), is via the relations y = − (3 x + x ) , y = 3 x ( x + x ) (23a)(see (12)) and their time derivatives,˙ y = − (3 ˙ x + ˙ x ) , ˙ y = 3 [(2 x + x ) ˙ x + x ˙ x ] . (cid:4) (23b) µ = µ = 2 In this case y = − x + x ) , y = ( x ) + ( x ) + 4 x x ,y = − x x ( x + x ) , y = ( x x ) . (24a) Remark 3.2-1 . Of course a remark completely analogous to Remark 3.1-1 holds in this case as well. (cid:4) The corresponding equations (8a) read˙ y ( x n ) + ˙ y ( x n ) + ˙ y x n + ˙ y = 0 , n = 1 , x = − (2 x + x ) ˙ y + ˙ y x − x ) , ˙ x = ( x + 2 x ) ˙ y + ˙ y x − x ) , (25a)˙ x = − x ( x + 2 x ) ˙ y − ˙ y h ( x ) − ( x ) i , ˙ x = x ( x + 2 x ) ˙ y + ˙ y h ( x ) − ( x ) i , (25b)˙ x = − ( x ) x ˙ y + ˙ y x x ( x − x ) , ˙ x = x ( x ) ˙ y + ˙ y x x ( x − x ) , (25c)˙ x = x ( x + 2 x ) ˙ y + (2 x + x ) ˙ y h ( x ) − ( x ) i , ˙ x = − x ( x + 2 x ) ˙ y + (2 x + x ) ˙ y h ( x ) − ( x ) i , (25d)˙ x = ( x ) x ˙ y − (2 x + x ) ˙ y x x h ( x ) − ( x ) i , ˙ x = − ( x ) x ˙ y − (2 x + x ) ˙ y x x h ( x ) − ( x ) i (25e)˙ x = − x x ˙ y + ( x + 2 x ) ˙ y x ( x ) ( x − x ) , ˙ x = x x ˙ y + ( x + 2 x ) ˙ y x ( x ) ( x − x ) . (25f)10ence, to the system (17), one now associates again the requirement (19);and—by making again the assignment (20a) for the system of evolution equa-tions satisfied by y ( t ) and y ( t )—one identifies again the restriction (20b),thereby concluding—via (25a)—that the polynomial system˙ x n = a + b h ( x n ) − x n x n +1 − x n +1 ) i , n = 1 , , (26)where now a = − β / b = 4 β / 9, is solvable . And the explicit solution isthen quite analogous (up to simple modifications of some parameters) to thatdescribed (after eq. (21)) in the preceding Subsection 3.1 . solvable systems of 2 nonlinearly-coupledODEs identified via the technique describedin Section 3 In this Section 4 we report a list of solvable systems of 2 nonlinearly coupledfirst-order ODEs satisfied by the 2 dependent variables x ( t ) and x ( t ); ineach case we identify the corresponding solvable system of 2 ODEs satisfiedby 2 variables y ˜ m ( t ) (for these, and other, notations used below see Section3 ); indeed, to help the reader mainly interested in the solvable character ofone of the following systems we also specify below on a case-by-case basis theinformation which allows to solve that specific system (we do so even at the costof minor repetitions). Note that the majority of these models feature equationsof type (4), but in a few cases the right-hand sides of these ODEs are not quitepolynomial . And let us recall that in this Section 4 parameters such as a, b, c (possibly equipped with indices) are arbitrary numbers (possibly complex ). Remark 4-1 . Most of the models reported below are characterized byevolution equations of the following kind:˙ x n = K X k =0 h p ( n ) k ( x , x ) i , n = 1 , , (27a)with K a positive integer and the functions p ( n ) k ( x , x ) homogenous polynomialsof degree k , p ( n ) k ( x , x ) = k X ℓ =0 h a ( n,k ) ℓ ( x ) k − ℓ ( x ) ℓ i , k = 0 , , ..., K , n = 1 , . (27b)So the different models are characterized by the assignments of the positiveinteger K and of the ( K + 1) parameters a ( n,k ) ℓ , expressed in each case interms of a few arbitrary parameters . It is of course obvious that in all the modelsassociated with Case (ii) (see Subsection 3.2 ) these parameters satisfy therestriction a (1 ,k ) ℓ = a (2 ,k ) ℓ , since in that case the 2 zeros x ( t ) and x ( t ) are11ompletely equivalent; while this restriction need not hold in Case (i) (see Subsection 3.1 ), although in some such cases it also emerges (see below). Itis on the other hand plain that, also in Case (i) (as, obviously, in Case (ii) ),there holds the restriction k X ℓ =0 h a (1 ,k ) ℓ i = k X ℓ =0 h a (2 ,k ) ℓ i , k = 0 , , ..., K , (27c)because for the special initial conditions x (0) = x (0)—implying x ( t ) = x ( t ) ≡ x ( t ), since in such case the distinction among Case (i) and Case(ii) obviously disappears—the 2 evolution equations (with n = 1 , 2) satisfied by x ( t ), ˙ x = x k k X ℓ =0 h a ( n,k ) ℓ i , k = 0 , , ..., K , n = 1 , , (27d)must coincide. (cid:4) Remark 4-2 . In the following 44 subsections we list as many solvable systems of 2 nonlinearly-coupled first-order ODEs, most of them with polyno-mial right-hand sides, and we indicate how each of them can be solved. Thepresentation of all these models is made so as to facilitate the utilization ofthese findings by practitioners only interested in one of these models (or itsgeneralization, see Section 5 ). Note however that not all these models are dif-ferent among themselves: indeed, some feature identical equations of motion—although the method to solve them might seem different. This is demonstratedby the following self-evident identification of the following equations of mo-tion: (32) ≡ (54), (33) ≡ (55), (40) ≡ (56) ≡ (70), (41) ≡ (57) ≡ (71), (48) ≡ (62) ≡ (68),(49) ≡ (63) ≡ (69). So in fact the list below contains only different systems of2 nonlinearly-coupled first-order differential equations for the 2 time-dependentvariables x ( t ) and x ( t ). (cid:4) ˙ x = x n a + b h 11 ( x ) + 6 x x − ( x ) io , (28a)˙ x = ax + b h − x ) + 9 ( x ) x + 12 x ( x ) + ( x ) i ; (28b) x ( t ) and x ( t ) are related to y ( t ) and y ( t ) by (12); and the variables y ( t )and y ( t ) evolve according to (82), the explicit solution of which is given bythe relevant formulas in Subsection Case A.1 of Appendix A with ˜ m = 1 , ˜ m = 2 , L = 1 , α = a, α = 3 b, β = 2 a, β = 16 b. ˙ x n = ax n + b h x n ) + 9 ( x n ) x n +1 − ( x n +1 ) i , n = 1 , ( t ) and x ( t ) are related to y ( t ) and y ( t ) by (24a); and the variables y ( t )and y ( t ) evolve according to (82), the explicit solution of which is given bythe relevant formulas in Subsection Case A.1 of Appendix A with ˜ m = 1 , ˜ m = 2 , L = 1 , α = a, α = (3 / b, β = 2 a, β = 4 b. ˙ x n = a + x n (cid:0) a + a X + a X (cid:1) + b X + b X + b X ,X ≡ x + x , n = 1 , x ( t ) and x ( t ) are related to y ( t ) and y ( t ) by (12); and the variables y ( t )and y ( t ) evolve according to (87), the explicit solution of which is given bythe relevant formulas in Subsection Case A.2 of Appendix A with ˜ m = 1 , ˜ m = 2 , L = 3 , α = − a , α = a + 4 b , α = − ( a + 4 b ) , α = a + 4 b ,β = 2 a , β = − a , β = 2 a , γ = − a , γ = 3 b , γ = − b , γ = 3 b . This model (30a) (with a = 0) is actually a special case of the more generalmodel˙ x n = a + L X ℓ =1 n ( − X ) ℓ − [ a ℓ x n + b ℓ X ] o , X ≡ x + x , n = 1 , , (30b)again with x ( t ) and x ( t ) related to y ( t ) and y ( t ) by (12) and the variables y ( t ) and y ( t ) evolving according to (87) with ˜ m = 1 , ˜ m = 2 , L an arbitrary positive integer, α = − a , α ℓ = a ℓ + 4 b ℓ , β ℓ = 2 a ℓ , γ = − a , γ ℓ = 3 b ℓ . ˙ x n = a + x n (cid:0) a + a X + a X (cid:1) + b X + b X + b X ,X ≡ x + x , n = 1 , x ( t ) and x ( t ) are related to y ( t ) and y ( t ) by (24a); and the variables y ( t )and y ( t ) evolve according to (87), the explicit solution of which is given bythe relevant formulas in Subsection Case A.2 of Appendix A with ˜ m = 1 , ˜ m = 2 , L = 3 , α = − a , α = a + 2 b , α = − a / − b , α = b / a / ,β = 2 a , β = − a , β = a / , γ = − a , γ = (3 / b , γ = − (3 / b ,γ = (3 / b . This model (31a) is actually a special case of the more general model˙ x n = a + L X ℓ =1 n ( − X ) ℓ − [ a ℓ x n + b ℓ X ] o , X ≡ x + x , n = 1 , , (31b)again with x ( t ) and x ( t ) related to y ( t ) and y ( t ) by (24a) and the variables y ( t ) and y ( t ) evolving according to (87) with ˜ m = 1 , ˜ m = 2 , L an arbitrary positive integer, α = − a , α ℓ = a ℓ + 2 b ℓ , β ℓ = 2 a ℓ , γ = − a , γ ℓ = (3 / b ℓ . .5 Model 4.(i)1.2c ˙ x n = x n (cid:2) a + bX + cX (cid:3) , X ≡ x ( x + x ) , n = 1 , x ( t ) and x ( t ) are related to y ( t ) and y ( t ) by (12); and the variables y ( t )and y ( t ) evolve according to (87), the explicit solution of which is given bythe relevant formulas in Subsection Case A.2 of Appendix A with ˜ m = 2 , ˜ m = 1 , L = 3 , α = 0 , α = 2 a, α = 2 b/ , α = 2 c/ , β = a, β = b/ ,β = c/ , γ ℓ = 0 . ˙ x n = x n (cid:0) a + bX + cX (cid:1) ,X ≡ ( x ) + 4 x x + ( x ) , n = 1 , x ( t ) and x ( t ) are related to y ( t ) and y ( t ) by (24a); and the variables y ( t )and y ( t ) evolve according to (87), the explicit solution of which is given bythe relevant formulas in Subsection Case A.2 of Appendix A with ˜ m = 2 , ˜ m = 1 , L = 3 , α = 0 , α = 2 a, α = 2 b, α = 2 c, β = a, β = b, β = c,γ ℓ = 0 . ˙ x = a + b h x ) + 10 x x + ( x ) i , (34a)˙ x = a + b h 17 ( x ) + 2 x x − x ) i ; (34b) x ( t ) and x ( t ) are related to y ( t ) and y ( t ) by (12); and the variables y ( t )and y ( t ) evolve according to (95a), the explicit solution of which is given by therelevant formulas in Subsection Case A.3.1 of Appendix A with ˜ m = 1 , ˜ m = 2 , α = − a, α = − (32 / b, β = − a, β = − b . Note that this is themodel treated in detail in Subsection 3.1.1 , see (21). ˙ x n = a + b h ( x n ) − x x − x n +1 ) i , n = 1 , x ( t ) and x ( t ) are related to y ( t ) and y ( t ) by (24a); and the variables y ( t )and y ( t ) evolve according to (95a), the explicit solution of which is given by therelevant formulas in Subsection Case A.3.1 of Appendix A with ˜ m = 1 , ˜ m = 2 , α = − a, α = 8 b, β = − a, β = (9 / b . Note that this is themodel treated in detail in Subsection 3.1.2 , see (26).14 .9 Model 4.(i)1.3a ˙ x = x n a + b h 65 ( x ) + 77 ( x ) x − x ( x ) − ( x ) io , (36a)˙ x = ax − b h 33 ( x ) + 15 ( x ) x − 147 ( x ) ( x ) − x ( x ) − x ) i ;(36b) x ( t ) and x ( t ) are related to y ( t ) and y ( t ) by (12); and the variables y ( t )and y ( t ) evolve according to (82), the explicit solution of which is given by therelevant formulas in Subsection Case A.1 of Appendix A ) with ˜ m = 1 , ˜ m = 3 , L = 1 , α = a, α = − b, β = 3 a, β = − b. ˙ x n = x n n a + b ( x + x ) h ( x n ) + 5 x x − x n +1 ) io , n = 1 , x ( t ) and x ( t ) are related to y ( t ) and y ( t ) by (24a); and the variables y ( t )and y ( t ) evolve according to (82), the explicit solution of which is given bythe relevant formulas in Subsection Case A.1 of Appendix A with ˜ m = 1 , ˜ m = 3 , L = 1 , α = a, α = − b/ , β = 3 a, β = − b. ˙ x = (6 x ) − h ( x ) (cid:0) a + a X + a X (cid:1) + (7 x + x ) (cid:0) b + b X + b X + b X (cid:1)(cid:3) , (38a)˙ x = (6 x ) − (cid:2) x x (cid:0) a + a X + a X (cid:1) + (11 x − x ) (cid:0) b + b X + b X + b X (cid:1)(cid:3) , (38b) X ≡ x + x ; (38c) x ( t ) and x ( t ) are related to y ( t ) and y ( t ) by (12); and the variables y ( t )and y ( t ) evolve according to (87), the explicit solution of which is givenby the relevant formulas in Subsection Case A.2 of Appendix A with˜ m = 1 , ˜ m = 3 , L = 3 , α = − (16 / b , α ℓ = ( − ℓ − ( a ℓ − / 6) + (16 / b ℓ ,β ℓ = ( − ℓ − a ℓ − / , ( ℓ = 1 , , , γ ℓ − = ( − ℓ b ℓ − , ( ℓ = 1 , , , . Note thatthe right-hand sides of these 2 ODEs, (38), are both polynomial only if the 4parameters b ℓ vanish, b ℓ = 0 , ℓ = 0 , , , .12 Model 4.(ii)1.3b ˙ x = 6 − (cid:2) x n (cid:0) a + a X + a X (cid:1) + ( x n + 3 x n +1 ) (cid:0) b X − + b + b X + b X (cid:1)(cid:3) ,X ≡ x + x , n = 1 , x ( t ) and x ( t ) are related to y ( t ) and y ( t ) by (24a); and the variables y ( t )and y ( t ) evolve according to (87), the explicit solution of which is given by therelevant formulas in Subsection Case A.2 of Appendix A with ˜ m = 1 , ˜ m =3 , L = 3 , α = − (16 / b , α ℓ = − ( − − ℓ (cid:0) a ℓ + 2 − ℓ b ℓ (cid:1) / , β ℓ = − ( − − ℓ − a ℓ ,γ = − b , γ ℓ = − ( − − ℓ b ℓ , ℓ = 1 , , . Note that the right-hand sides of these2 ODEs, (38), are both polynomial iff the single parameter b vanishes, b = 0. ˙ x n = x n (cid:0) a + bX + cX (cid:1) , X ≡ ( x ) ( x + 3 x ) , n = 1 , x ( t ) and x ( t ) are related to y ( t ) and y ( t ) by (12); and the variables y ( t )and y ( t ) evolve according to (87), the explicit solution of which is given bythe relevant formulas in Subsection Case A.2 of Appendix A with ˜ m = 3 , ˜ m = 1 , L = 3 , α = 0 , α = 3 a, α = − b, α = 3 c, β = a, β = − b, β = c,γ ℓ = 0 . ˙ x n = x n (cid:0) a + bX + cX (cid:1) , X ≡ x x ( x + x ) , n = 1 , x ( t ) and x ( t ) are related to y ( t ) and y ( t ) by (24a); and the variables y ( t )and y ( t ) evolve according to (87), the explicit solution of which is given by therelevant formulas in Subsection Case A.2 of Appendix A ) with ˜ m = 3 , ˜ m = 1 , L = 3 , α = 0 , α = 3 a, α = − b/ , α = 3 c/ , β = a, β = − b/ ,β = c/ , γ ℓ = 0 . ˙ x = (6 x ) − { a (7 x + x )+ b h 13 ( x ) + 376 ( x ) x + 106 ( x x ) + 16 x ( x ) + ( x ) io , (42a)˙ x = (6 x ) − { a (11 x − x )+ b h 473 ( x ) + 408 ( x ) x − 318 ( x x ) − x ( x ) − x ) io ; (42b)16 ( t ) and x ( t ) are related to y ( t ) and y ( t ) by (12); and the variables y ( t )and y ( t ) evolve according to (95a), the explicit solution of which is given by therelevant formulas in Subsection Case A.3.2 of Appendix A with ˜ m = 1 , ˜ m = 3 , α = − (16 / a, α = (cid:0) / (cid:1) b, β = − a, β = b . Note that theright-hand sides of the 2 ODEs (42) are not polynomial. ˙ x = a (cid:18) x n + 3 x n +1 x + x (cid:19) + b h x n ) − ( x n ) x n +1 − x n ( x n +1 ) − x n +1 ) i ,n = 1 , x ( t ) and x ( t ) are related to y ( t ) and y ( t ) by (24a); and the variables y ( t )and y ( t ) evolve according to (95a), the explicit solution of which is given by therelevant formulas in Subsection Case A.3.2 of Appendix A with ˜ m = 1 , ˜ m = 3 , α = − a, α = − b, β = − (3 / a, β = − (3 / b . Note that theright-hand sides of the 2 ODEs (43) are polynomial only if a = 0 . ˙ x = x { a + ( b/ ·· h 243 ( x ) + 648 ( x ) x − 106 ( x x ) − x ( x ) − ( x ) io , (44a)˙ x = x { a − b ·· h 243 ( x ) − 376 ( x ) x − 106 ( x x ) − x ( x ) − ( x ) io ; (44b) x ( t ) and x ( t ) are related to y ( t ) and y ( t ) by (12); and the variables y ( t )and y ( t ) evolve according to (82), the explicit solution of which is given bythe relevant formulas in Subsection Case A.1 of Appendix A with ˜ m = 1 , ˜ m = 4 , L = 1 , α = a, α = b, β = 4 a, β = 2 b = 1024 b . ˙ x n = x n { a + b ·· h ( x n ) + 6 ( x n ) x n +1 + 16 ( x x ) − x n ( x n +1 ) − ( x n +1 ) io ,n = 1 , , (45)17 ( t ) and x ( t ) are related to y ( t ) and y ( t ) by (24a); and the variables y ( t )and y ( t ) evolve according to (82), the explicit solution of which is given bythe relevant formulas in Subsection Case A.1 of Appendix A with ˜ m = 1 , ˜ m = 4 , L = 1 , α = a, α = b/ , β = 4 a, β = 2 b = 64 b . ˙ x = x (cid:0) a + a X + a X (cid:1) − " 37 ( x ) + 10 x x + ( x ) x ) b + b X + b X + b X (cid:1) , (46a)˙ x = x (cid:0) a + a X + a X (cid:1) − " 27 ( x ) − x x − ( x ) ( x ) b + b X + b X + b X (cid:1) ; (46b) X ≡ x + x ; (46c) x ( t ) and x ( t ) are related to y ( t ) and y ( t ) by (12); and the variables y ( t )and y ( t ) evolve according to (87), the explicit solution of which is given bythe relevant formulas in Subsection Case A.2 of Appendix A with ˜ m = 1 , ˜ m = 4 , L = 3 , α = 64 b ; α ℓ = ( − ℓ − ( a ℓ − − b ℓ ) , β ℓ = ( − ℓ − a ℓ − ,ℓ = 1 , , γ ℓ = ( − ℓ b ℓ , ℓ = 0 , , , . Note that the right-hand sides of these 2ODEs, (46), are both polynomial only if all the 4 parameters b ℓ vanish, b ℓ = 0 ,ℓ = 0 , , , ˙ x n = x n (cid:0) a + a X + a X (cid:1) + " ( x n ) − x x − ( x n +1 ) x x ·· (cid:0) b + b X + b X + b X (cid:1) , X ≡ x + x , n = 1 , x ( t ) and x ( t ) are related to y ( t ) and y ( t ) by (24a); and the variables y ( t )and y ( t ) evolve according to (87), the explicit solution of which is given bythe relevant formulas in Subsection Case A.2 of Appendix A with ˜ m = 1 , ˜ m = 4 , L = 3 , α = 16 b ; α ℓ = ( − − ℓ a ℓ − + ( − − ℓ b ℓ , β ℓ = ( − − ℓ a ℓ − ,ℓ = 1 , , γ ℓ = ( − − − ℓ b ℓ , ℓ = 0 , , , . Note that the right-hand sides ofthese 2 ODEs, (47), are both polynomial only if all the 4 parameters b ℓ vanish, b ℓ = 0 , ℓ = 1 , , 3. 18 .21 Model 4.(i)1.4c ˙ x n = x n (cid:0) a + bX + cX (cid:1) , X ≡ ( x ) x , n = 1 , x ( t ) and x ( t ) are related to y ( t ) and y ( t ) by (12); and the variables y ( t )and y ( t ) evolve according to (87), the explicit solution of which is given bythe relevant formulas in Subsection Case A.2 of Appendix A with ˜ m = 4 , ˜ m = 1 , L = 3 , α = 0 , α = 4 a, α = 4 b, α = 4 c, β = a, β = b, β = c,γ ℓ = 0 . ˙ x n = x n (cid:0) a + bX + cX (cid:1) , X ≡ ( x x ) , n = 1 , x ( t ) and x ( t ) are related to y ( t ) and y ( t ) by (24a); and the variables y ( t )and y ( t ) evolve according to (87), the explicit solution of which is given bythe relevant formulas in Subsection Case A.2 of Appendix A with ˜ m = 4 , ˜ m = 1 , L = 3 , α = 0 , α = 4 a, α = 4 b, α = 4 c, β = a, β = b, β = c,γ ℓ = 0 . ˙ x = h x ) i − n a h 37 ( x ) + 10 x x + ( x ) i + b h x ) − x ) x − x ) ( x ) − x x ) − 211 ( x ) ( x ) − x ( x ) − ( x ) io , (50a)˙ x = h ( x ) i − n a h 27 ( x ) − x x − ( x ) i − b h x ) + 7290 ( x ) x − x ) ( x ) − x x ) − 211 ( x ) ( x ) − x ( x ) − ( x ) io ; (50b) x ( t ) and x ( t ) are related to y ( t ) and y ( t ) by (12); and the variables y ( t )and y ( t ) evolve according to (95a), the explicit solution of which is given by therelevant formulas in Subsection Case A.3.3 of Appendix A with ˜ m = 1 , ˜ m = 4 , α = − a = − a ; α = 2 b = 1638 b, β = − a, β = b. Note thatthe right-hand sides of these 2 ODEs, (50), are not polynomial .19 .24 Model 4.(ii)1.4d ˙ x n = ( x x ) − n a h ( x n ) − x x − ( x n +1 ) i + b h ( x n ) + 8 ( x n ) x n +1 + 29 ( x n ) ( x n +1 ) − 64 ( x x ) − 29 ( x n ) ( x n +1 ) − x n ( x n +1 ) − ( x n +1 ) io ,n = 1 , x ( t ) and x ( t ) are related to y ( t ) and y ( t ) by (24a); and the variables y ( t )and y ( t ) evolve according to (95a), the explicit solution of which is given by therelevant formulas in Subsection Case A.3.3 of Appendix A with ˜ m = 1 , ˜ m = 4 , α = 2 a = 16 a ; α = 2 b = 256 b, β = 2 − a = a/ , β = 2 − b = b/ . Note that the right-hand sides of these 2 ODEs, (50), are not polynomial . ˙ x = x n a + b ( x ) h x ) + 10 ( x ) x + 7 x ( x ) − x ) io , (52a)˙ x = ax − b ( x ) h x ) + 7 ( x ) x − x ( x ) − x ) i ; (52b) x ( t ) and x ( t ) are related to y ( t ) and y ( t ) by (12); and the variables y ( t )and y ( t ) evolve according to (82), the explicit solution of which is given bythe relevant formulas in Subsection Case A.1 of Appendix A with ˜ m = 2 , ˜ m = 3 , L = 1 , α = 2 a, α = (4 / b, β = 3 a, β = 3 b. ˙ x n = x n (cid:26) a + b h ( x ) + x x + ( x ) i − ·· h ( x n ) + 19 ( x n ) x n +1 + 151 ( x n ) ( x n +1 ) + 331 ( x n ) ( x n +1 ) +259 ( x x ) + 13 ( x n ) ( x n +1 ) − 89 ( x n ) ( x n +1 ) − x n ( x n +1 ) − x n +1 ) io , n = 1 , x ( t ) and x ( t ) are related to y ( t ) and y ( t ) by (24a); and the variables y ( t )and y ( t ) evolve according to (82), the explicit solution of which is given bythe relevant formulas in Subsection Case A.1 of Appendix A with ˜ m = 2 , ˜ m = 3 , L = 1 , α = 2 a, α = 2 b, β = 3 a, β = (81 / b. Note that theright-hand sides of these ODEs is not polynomial.20 .27 Model 4.(i)2.3b ˙ x n = x n (cid:0) a + bX + cX (cid:1) , X ≡ x ( x + x ) , n = 1 , x ( t ) and x ( t ) are related to y ( t ) and y ( t ) by (12); and the variables y ( t )and y ( t ) evolve according to (87), the explicit solution of which is given bythe relevant formulas in Subsection Case A.2 of Appendix A with ˜ m = 2 , ˜ m = 3 , L = 3 , α = 0 , α = 2 a, α = (2 / b, α = (2 / c, β = 3 a,β = b, β = c/ , γ ℓ = 0 . ˙ x n = x n (cid:0) a + bX + cX (cid:1) , X ≡ ( x ) + 4 x x + ( x ) , n = 1 , x ( t ) and x ( t ) are related to y ( t ) and y ( t ) by (24a); and the variables y ( t )and y ( t ) evolve according to (87), the explicit solution of which is given bythe relevant formulas in Subsection Case A.2 of Appendix A with ˜ m = 2 , ˜ m = 3 , L = 3 , α = 0 , α = 2 a, α = 2 b, α = 2 c, β = 3 a, β = 3 b, β =3 c, γ ℓ = 0 . ˙ x n = x n (cid:0) a + bX + cX (cid:1) , X ≡ ( x ) ( x + 3 x ) , n = 1 , x ( t ) and x ( t ) are related to y ( t ) and y ( t ) by (12); and the variables y ( t )and y ( t ) evolve according to (87), the explicit solution of which is given bythe relevant formulas in Subsection Case A.2 of Appendix A with ˜ m = 3 , ˜ m = 2 , L = 3 , α = 0 , α = 3 a, α = − b, α = 3 c, β = 2 a, β = − b,β = 3 c, γ ℓ = 0 . ˙ x n = x n (cid:0) a + bX + cX (cid:1) , X ≡ x x ( x + x ) , n = 1 , x ( t ) and x ( t ) are related to y ( t ) and y ( t ) by (24a); and the variables y ( t )and y ( t ) evolve according to (87), the explicit solution of which is given bythe relevant formulas in Subsection Case A.2 of Appendix A with ˜ m = 3 , ˜ m = 2 , L = 3 , α = 0 , α = 3 a, α = − (3 / b, α = (3 / c, β = 2 a,β = − b, β = c/ , γ ℓ = 0 . .31 Model 4.(i)2.4a ˙ x = x n a + b ( x ) h ( x ) + 4 x x − ( x ) i + c ( x ) h ( x ) + 6 ( x ) x + 16 ( x x ) − x ( x ) − ( x ) io , (58a)˙ x = x n a − b ( x ) h x ) − x x − x ) i + c ( x ) h − x ) − 18 ( x ) x + 16 ( x x ) + 18 x ( x ) + 3 ( x ) io , (58b) x ( t ) and x ( t ) are related to y ( t ) and y ( t ) by (12); and the variables y ( t )and y ( t ) evolve according to (82), the explicit solution of which is given bythe relevant formulas in Subsection Case A.1 of Appendix A with ˜ m = 2 , ˜ m = 4 , L = 2 , α = 2 a, α = (2 / b, α = (2 / c, β = 4 a, β = 16 b,β = 64 c. ˙ x n = x n n a + ( x + x ) − · n b h ( x n ) + 13 ( x n ) x n +1 + 64 ( x n ) ( x n +1 ) +8 ( x n ) ( x n +1 ) − x n ( x n +1 ) − ( x n +1 ) i + c h ( x n ) + 21 ( x n ) x n +1 + 186 ( x n ) ( x n +1 ) + 906 ( x n ) ( x n +1 ) +2676 ( x n ) ( x n +1 ) − 84 ( x n ) ( x n +1 ) − 906 ( x n ) ( x n +1 ) − 186 ( x n ) ( x n +1 ) − x n ( x n +1 ) − ( x n +1 ) ioo ,n = 1 , x ( t ) and x ( t ) are related to y ( t ) and y ( t ) by (24a); and the variables y ( t )and y ( t ) evolve according to (82), the explicit solution of which is given bythe relevant formulas in Subsection Case A.1 of Appendix A with ˜ m =2 , ˜ m = 4 , L = 2 , α = 2 a, α = 2 b, α = − c, β = 4 a, β = 144 b,β = − c = − c. Note that the right-hand sides of these ODEs are not polynomial, except for the trivial case with b = c = 0 . ˙ x = x (cid:0) a + a X + a X (cid:1) + ( x ) − (cid:0) b + b X + b X + b X (cid:1) , (60a)˙ x = x (cid:0) a + a X + a X (cid:1) + (2 x − x )( x ) (cid:0) b + b X + b X + b X (cid:1) , (60b)22 ≡ x ( x + x ) ; (60c) x ( t ) and x ( t ) are related to y ( t ) and y ( t ) by (12); and the variables y ( t )and y ( t ) evolve according to (87), the explicit solution of which is given bythe relevant formulas in Subsection Case A.2 of Appendix A with ˜ m = 2 , ˜ m = 4 , L = 3 , α = 12 b ; α ℓ = 2 (cid:0) − ℓ (cid:1) a ℓ − + 4 (cid:0) − ℓ (cid:1) b ℓ , β ℓ = 4 (cid:0) − ℓ (cid:1) a ℓ − ,ℓ = 1 , , γ ℓ = 2 (cid:0) − − ℓ (cid:1) b ℓ , ℓ = 0 , , , . Note that the right-hand sides ofthese 2 ODEs, (46), are both polynomial only if all the 4 parameters b ℓ vanish, b ℓ = 0 , ℓ = 0 , , , ˙ x n = x n (cid:8) a + a X + a X + " − x n ) + 7 x n x n +1 + ( x n +1 ) x x ( x + x ) b + b X + b X + b X (cid:1)) ,X ≡ ( x ) + 4 x x + ( x ) ; (61) x ( t ) and x ( t ) are related to y ( t ) and y ( t ) by (24a); and the variables y ( t )and y ( t ) evolve according to (87), the explicit solution of which is given bythe relevant formulas in Subsection Case A.2 of Appendix A with ˜ m = 2 , ˜ m = 4 , L = 3 , α = 36 b ; α ℓ = 2 a ℓ − + 36 b ℓ , β ℓ = 4 a ℓ − , ℓ = 1 , , γ ℓ = 2 b ℓ ,ℓ = 0 , , , . Note that the right-hand sides of these 2 ODEs, (61), are both polynomial only if all the 4 parameters b ℓ vanish, b ℓ = 0 , ℓ = 0 , , , ˙ x n = x n (cid:0) a + bX + cX (cid:1) , X ≡ ( x ) x , n = 1 , x ( t ) and x ( t ) are related to y ( t ) and y ( t ) by (12); and the variables y ( t )and y ( t ) evolve according to (87), the explicit solution of which is given bythe relevant formulas in Subsection Case A.2 of Appendix A with ˜ m = 4 , ˜ m = 2 , L = 3 , α = 0 , α = 4 a, α = 4 b, α = 4 c, β = 2 a, β = 2 b, β =2 c, γ ℓ = 0 . ˙ x n = x n (cid:0) a + bX + cX (cid:1) , X ≡ ( x x ) , n = 1 , x ( t ) and x ( t ) are related to y ( t ) and y ( t ) by (24a); and the variables y ( t )and y ( t ) evolve according to (87), the explicit solution of which is given bythe relevant formulas in Subsection Case A.2 of Appendix A with ˜ m = 4 , ˜ m = 2 , L = 3 , α = 0 , α = 4 a, α = 4 b, α = 4 c, β = 2 a, β = 2 b, β =2 c, γ ℓ = 0 . .37 Model 4.(i)2.4d ˙ x = ( x ) − n a + b ( x ) h ( x ) − x x − ( x ) io , (64a)˙ x = ( x ) − n a (2 x − x ) − b ( x ) h x ) + 9 ( x ) x − x ( x ) − ( x ) io , (64b) x ( t ) and x ( t ) are related to y ( t ) and y ( t ) by (12); and the variables y ( t )and y ( t ) evolve according to (95a), the explicit solution of which is given by therelevant formulas in Subsection Case A.3.1 of Appendix A with ˜ m = 2 , ˜ m = 4 , α = 12 a ; α = − b, β = (2 / a, β = − (2 / b. Note that theright-hand sides of these 2 ODEs, (50), are not polynomial , unless a vanishes. ˙ x n = [ x x ( x + x )] − n a h x n ) − x x − ( x n +1 ) i + b h x n ) + 27 ( x n ) x n +1 + 141 ( x n ) ( x n +1 ) − 280 ( x x ) − 90 ( x n ) ( x n +1 ) − x n ( x n +1 ) − ( x n +1 ) io ,n = 1 , x ( t ) and x ( t ) are related to y ( t ) and y ( t ) by (24a); and the variables y ( t )and y ( t ) evolve according to (95a), the explicit solution of which is given by therelevant formulas in Subsection Case A.3.1 of Appendix A with ˜ m = 2 , ˜ m = 4 , α = − a ; α = − b, β = − a, β = − b. Note that the right-hand sides of these 2 ODEs, (65), are not polynomial. ˙ x = x n a + b ( x ) ·· h ( x ) + 16 ( x ) x + 106 ( x x ) + 376 x ( x ) − 243 ( x ) io , (66a)˙ x = x n a − b ( x ) ·· h x ) + 48 ( x ) x + 318 ( x x ) + 104 x ( x ) − 729 ( x ) io ; (66b) x ( t ) and x ( t ) are related to y ( t ) and y ( t ) by (12); and the variables y ( t )and y ( t ) evolve according to (82), the explicit solution of which is given bythe relevant formulas in Subsection Case A.1 of Appendix A with ˜ m = 3 , ˜ m = 4 , L = 1 , α = 3 a, α = 3 b, β = 4 a, β = 2 b = 1024 b. .40 Model 4.(ii)3.4a ˙ x n = x n n a + b ( x x ) ·· h x n ) + 18 ( x n ) x n +1 + 16 ( x x ) − x n ( x n +1 ) − x n +1 ) io ,n = 1 , x ( t ) and x ( t ) are related to y ( t ) and y ( t ) by (24a); and the variables y ( t )and y ( t ) evolve according to (82), the explicit solution of which is given bythe relevant formulas in Subsection Case A.1 of Appendix A with ˜ m = 3 , ˜ m = 4 , L = 1 , α = 3 a, α = (3 / b, β = 4 a, β = 64 b. ˙ x n = x n (cid:0) a + bX + cX (cid:1) , X ≡ ( x ) x , n = 1 , x ( t ) and x ( t ) are related to y ( t ) and y ( t ) by (12); and the variables y ( t )and y ( t ) evolve according to (87), the explicit solution of which is given bythe relevant formulas in Subsection Case A.2 of Appendix A with ˜ m = 4 , ˜ m = 3 , L = 3 , α = 0 , α = 4 a, α = 4 b, α = 4 c, β = 3 a, β = 3 b, β =3 c, γ ℓ = 0 . ˙ x n = x n (cid:2) a + bX + cX (cid:3) , X = ( x x ) n = 1 , x ( t ) and x ( t ) are related to y ( t ) and y ( t ) by (24a); and the variables y ( t )and y ( t ) evolve according to (87), the explicit solution of which is given bythe relevant formulas in Subsection Case A.2 of Appendix A with ˜ m = 4 , ˜ m = 3 , L = 3 , α = 0 , α = 4 a, α = 4 b, α = 4 c, β = 3 a, β = 3 b, β = 3 c,γ ℓ = 0 . ˙ x n = x n (cid:2) a + bX + cX (cid:3) , X ≡ ( x ) ( x + 3 x ) , n = 1 , x ( t ) and x ( t ) are related to y ( t ) and y ( t ) by (12); and the variables y ( t )and y ( t ) evolve according to (87), the explicit solution of which is given bythe relevant formulas in Subsection Case A.2 of Appendix A with ˜ m = 3 , ˜ m = 4 , L = 3 , α = 0 , α = 3 a, α = − b, α = 3 c, β = 4 a, β = − b, β =4 c, γ ℓ = 0 . .44 Model 4.(ii)3.4c ˙ x n = x n (cid:0) a + bX + cX (cid:1) , X ≡ x x ( x + x ) n = 1 , x ( t ) and x ( t ) are related to y ( t ) and y ( t ) by (24a); and the variables y ( t )and y ( t ) evolve according to (87), the explicit solution of which is given bythe relevant formulas in Subsection Case A.2 of Appendix A with ˜ m = 3 , ˜ m = 4 , L = 3 , α = 0 , α = 3 a, α = − (3 / b, α = (3 / c, β = 4 a,β = − b, β = c, γ ℓ = 0 . In this Section 5 we tersely indicate the possibility to generalize the class of solvable models listed in the preceding Section 4 , by outlining the procedureto do so in just one case, that detailed in Subsection 4.3 (see (30a)), in factjust the special case of it with a = a = a = b = b = 0, so that its equationsof motion read as follows:˙ x n = ( a x n + b X ) X, X ≡ x + x , n = 1 , , (72a)namely ˙ x n = c n ( x ) + c n ( x ) + c n x x , n = 1 , , (72b)with the 6 parameters c nm , n = 1 , , m = 1 , , 3, expressed as follows in termsof the 2 a priori arbitrary parameters a and b (see (30a)): c = 3 ( a + 3 b ) , c = b , c = a + 6 b ,c = 9 b , c = a + b , c = 3 ( a + 2 b ) . (72c)The explicit solution of the initial-values problem of this system (72) is providedby Remark A.2-1 (see Subsection A.2 of Appendix A ). Remark 5-1 . Note that the right-hand sides of the 2 ODEs (72) are homo-geneous polynomials of second degree , the coefficients of which satisfy of coursethe condition X ℓ =1 ( c ℓ ) = X ℓ =1 ( c ℓ ) , (73)as implied by Remark 4-1 . Moreover—as clearly implied by (72a)—the 2homogeneous second-degree polynomials in the right-hand sides of the 2 ODEscharacterizing this model feature a common zero: they both vanish when X = 0,namely when x = − x . (cid:4) Analogous extensions of other models treated in this paper shall be per-formed by practitioners interested in these systems of ODEs in the context ofspecific applications (see Section 6 ). 26 emark 5-2 . Note that, via a by now well-known trick (see, for instance,[7]) corresponding to the following time-dependent change of both independentand dependent variables, τ = exp ( at ) ; X n ( t ) = exp ( at ) x n ( τ ) , n = 1 , , (74)the autonomous system (72) gets replaced by the following, also autonomous ,system: ˙ X n = aX n + c n ( X ) + c n ( X ) + c n X X , n = 1 , . (75)Here a is an arbitrary (time-independent) parameter; and note that if this pa-rameter a is purely imaginary , Re [ a ] = 0 , Im [ a ] = 0 , then this dynamical system(75) is generally doubly periodic ; or even—if a /b is a real rational number— isochronous , namely then all its solutions are completely periodic with a period(an integer multiple of T = 2 π/ | a | ) independent of the initial data: see RemarkA.2-1 and, if need be, [7] [3]. (cid:4) Because of this remarkable fact, in the remaining part of this Section 5 we limit, for simplicity, consideration to the special case (72), by investigatingits extension which obtains via the following linear reshuffle of the 2 dependentvariables x ( t ) and x ( t ): z = A x + A x , z = A x + A x , (76a)which is inverted to read as follows x = ( A z − A z ) /D , x = ( − A z + A z ) /D ; (76b)here and hereafter D = A A − A A . (76c)It is easily seen that the new system then reads˙ z n = a n ( z ) + a n ( z ) + a n z z , n = 1 , , (77)with the 6 parameters a nℓ , n = 1 , , ℓ = 1 , , arbitrary parameters A nm , n = 1 , , m = 1 , , and the 2 arbitrary parameters a and b (see (72c)) as follows: a n = D − h ( A ) ( A n c + A n c ) + ( A ) ( A n c + A n c ) − A A ( A n c + A n c )] , n = 1 , , (78a) a n = D − h ( A ) ( A n c + A n c ) + ( A ) ( A n c + A n c ) − A A ( A n c + A n c )] , n = 1 , , (78b) a n = D − [ − A A ( A n c + A n c ) − A A ( A n c + A n c )+ ( A A + A A ) ( A n c + A n c )] , n = 1 , . (78c)27 emark 5.3 . The fact that the 6 parameters a nℓ which characterize thesystem (77) can be ( explicitly !) expressed, see (78), in terms of 6 a prioriarbitrary parameters—the 4 parameters A nm , see (76), and the 2 parameters a and b (see (72))—might seem to imply that this system (77) can be reducedby algebraic operations to the algebraically solvable system (72)— hence thatit is itself algebraically solvable— for any generic assignment of its 6 parameters a nℓ , n = 1 , , ℓ = 1 , , 3. That this is not the case is however implied by theobservation that the property of the system (72)—to feature in the right-handsides of its 2 ODEs 2 polynomials themselves featuring a common zero (see Remark 5-1 )—is then clearly also featured by the generalized system (77) (welike to thank Fran¸cois Leyvraz for this very useful observation). Hence only (atmost) 5 of the 6 parameters a nℓ ( n = 1 , ℓ = 1 , , 3) can be arbitrarily assigned,since these 6 parameters are constrained by the condition( a a − a a ) + ( a a − a a ) ( a a − a a ) = 0 (79)which is easily seen to correspond to the requirement that the right-hand sidesof the 2 ODEs (77) (with n = 1 , 2) feature a common zero. (cid:4) Remark 5-4 . Let us finally emphasize that the trick reported in Remark5-1 is just as applicable to the more general system (77), implying—via the ansatz τ = exp ( at ) ; Z n ( t ) = exp ( at ) z n ( τ ) , n = 1 , , (80a)analogous to (74)—the solvability of the system˙ Z n = aZ n + a n ( Z ) + a n ( Z ) + a n Z Z , n = 1 , , (80b)featuring the 7 parameters a and a nℓ ( n = 1 , ℓ = 1 , , (cid:4) The relevance of this dynamical system, (80b), in many applicative contextis exemplified by too many contributions to allow reporting a full bibliography;we record here just one such paper which lists 11 references and contains theremarkable assertion that the system (80b) ”is not solvable explicitly except incertain simple cases” [13]. In this final Section 6 we tersely outline future developments of the findingsreported in this paper.There is of course the possibility to treat cases with M > Section 3 ).There is the possibility to iterate the procedure leading to the identifica-tion of new solvable systems (as described in this paper): see for this kind ofdevelopment [14] and Chapter 6 of [7].Another natural development is to treat analogous dynamical systems evolv-ing in discrete rather than continuous time. For progress in this direction see[15]. 28nother extension is to treat systems characterized by second-order ratherthan first-order differential equations, including models characterized by New-tonian equations of motion (”accelerations equal forces”); and in the cases inwhich these equations of motion are derivable from a Hamiltonian, an additionalinteresting development is the treatment of the corresponding time-evolutionsin the context of quantal rather than classical mechanics.And yet another extension is to Partial Differential Equations (PDEs) ratherthan ODEs.There is finally the vast universe of applications, including to cases in whichthe systems of evolution equations can be shown—via their solvability —to fea-ture remarkable properties such as isochrony [2] [3] or asymptotic isochrony [4]. FP likes to thank the Physics Department of the University of Rome ”LaSapienza” for the hospitality from April to November 2018 (during her sabbat-ical), when the results reported in this paper were obtained. FC likes to thankRobert Conte and Fran¸cois Leyvraz for very useful discussions in the context ofthe Gathering of Scientists on ”Integrable systems and beyond” hosted by theCentro Internacional de Ciencias (CIC) in Cuernavaca, Mexico, from November19th to December 14th, 2018. nonlinear first-order ODEs forthe variables y ˜ m ( t ) The findings reported in this Appendix A are not new; they are displayed hereto facilitate the reader of the new findings reported in the body of this paper. Notation A-1 . In this Appendix A we indicate with the notation y ˜ m ( t )and y ˜ m ( t )—with ˜ m , = 1 , , , m = ˜ m (and for the significance of thesuperimposed tilde see the last part of Section 2 )—the 2 dependent variableswhich satisfy the ”solvable” system of 2 nonlinearly-coupled ODEs˙ y ˜ m = f ˜ m ( y ˜ m , y ˜ m ) , ˙ y ˜ m = f ˜ m ( y ˜ m , y ˜ m ) , (81)with the 2 functions f ˜ m ( y ˜ m , y ˜ m ) and f ˜ m ( y ˜ m , y ˜ m ) assigned—convenientlyfor our treatment in this paper (see Section 2 above)—so that the system (81)is ”solvable”. The precise meaning of the term ”solvable” shall be clear fromthe following.In this Appendix A α ℓ , β ℓ , γ ℓ are a priori arbitrary time-independent parameters, and L is an a priori arbitrary nonnegative integer .29he selection of the specific systems of 2 ODEs considered below is of coursemotivated by the treatment in the body of this paper, see in particular Sections2 and . (cid:4) ˙ y ˜ m = L X ℓ =0 h α ℓ ( y ˜ m ) ℓ ˜ m +1 i , ˙ y ˜ m = L X ℓ =0 h β ℓ ( y ˜ m ) ℓ ˜ m +1 i . (82)Each of these 2 ODEs can be integrated via one quadrature , that can be per-formed explicitly after some purely algebraic operations. Indeed, to integrate the first of these 2 ODEs one must first of all identify—via an algebraic operation—the L ˜ m + 1 zeros ¯ y n (assumed below, for simplicity, to be all different amongthemselves) of the polynomial in its right-hand side, L X ℓ =0 h α ℓ ( y ˜ m ) ℓ ˜ m +1 i = α L L ˜ m +1 Y n =1 ( y ˜ m − ¯ y n ) ; (83)next one must identify the L ˜ m +1 ”residues” r n defined by the ”partial fractiondecomposition” formula L ˜ m +1 Y n =1 ( y ˜ m − ¯ y n ) − = L ˜ m +1 X n =1 h r n ( y ˜ m − ¯ y n ) − i (84)—another algebraic operation, indeed one that can be performed explicitly ; andfinally one integrates the ODE getting the (generally implicit; but not always,see below) result L ˜ m +1 Y n =1 (cid:26)(cid:20) y ˜ m ( t ) − ¯ y n y ˜ m (0) − ¯ y n (cid:21) r n (cid:27) = exp (cid:18) tα L (cid:19) , (85)which characterizes the solution y ˜ m ( t ) corresponding to the initial datum y ˜ m (0) . Of course an analogous procedure characterizes—for the second ODE (82)—the solution y ˜ m ( t ) corresponding to the initial datum y ˜ m (0) . For L = 1 the initial-value problem for these 2 ODEs can be solved explicitly ,since the solution of the initial-value problem for the ODE˙ y = Ay + By M +1 (86a)is provided by the formula y ( t ) = y (0) exp ( At ) n B/A ) [ y (0)] M [1 − exp ( M At )] o − . (86b)30 .2 Case A.2 ˙ y ˜ m = L X ℓ =0 h α ℓ ( y ˜ m ) ℓ i , (87a)˙ y ˜ m = y ˜ m L X ℓ =1 h β ℓ ( y ˜ m ) ℓ − i + L X ℓ =0 h γ ℓ ( y ˜ m ) ℓ − m / ˜ m ) i . (87b)The solution of the first of these 2 ODEs, (87a), has been already discussedabove, see Subsection Case A.1 ; hence in this Subsection Case A.2 we needto consider only the second ODE (87b). And since we are mainly interested inthe case when the right-hand side of this ODE is polynomial, we shall limit ourconsideration below to the 3 subcases with ˜ m = 1 and to the single case with˜ m = 2 , ˜ m = 4; except in the special case with all parameters γ ℓ vanishing, γ ℓ = 0, which we treat separately firstly (since it is an intermediate step to solvethe more general case).In this special case the ODE (87b) reads˙ y ˜ m = y ˜ m L X ℓ =1 h β ℓ ( y ˜ m ) ℓ − i , (88a)with the function y ˜ m ( t ) to be considered known; hence the solution of theinitial-value problem for this ODE reads y ˜ m ( t ) = y ˜ m (0) F ( t ) (88b)with F ( t ) = exp (Z t " dt ′ L X ℓ =1 n β ℓ [ y ˜ m ( t ′ )] ℓ − o . (88c)And it is then easily seen that the solution of the initial-value problem ofthe (more general) ODE (87b) reads y ˜ m ( t ) = F ( t ) " y ˜ m (0) + Z t dt ′ [ F ( t ′ )] − L X ℓ =0 n γ ℓ [ y ˜ m ( t ′ )] ℓ − m / ˜ m ) o . (89)More explicit solutions can be easily obtained in the following cases:˜ m = 1 , ˜ m = 2 , , m = 2 , ˜ m = 4 , M = ˜ m / ˜ m , (90a)˙ y ˜ m = α + α y ˜ m + α ( y ˜ m ) , (90b)˙ y ˜ m = y ˜ m ( β + β y ˜ m ) + γ ( y ˜ m ) − M + γ ( y ˜ m ) M + γ ( y ˜ m ) M ; (90c) y ˜ m ( t ) = y ˜ m (0) [1 + (∆ /α ) tanh (∆ t )] − α /α ) tanh (∆ t )1 − (cid:8)(cid:2) α y ˜ m (0) + ∆ (cid:3) /α (cid:9) tanh (∆ t ) , ∆ = ( α ) − α α , (90d)31 ˜ m ( t ) = f ( t ) " y ˜ m (0) + Z t dt ′ [ f ( t ′ )] − L X ℓ =0 n γ ℓ [ y ˜ m ( t ′ )] ℓ − m / ˜ m ) o , (91a) f ( t ) = exp (Z t " dt ′ L X ℓ =1 n β ℓ [ y ˜ m ( t ′ )] ℓ − o . (91b) Remark A.2-1 . In particular, it is easily seen that the system of ODEs (72)discussed in Section 5 (see Subsection 4.3 with a = a = a = b = b = 0)implies that the system of ODEs (87), which then reads˙ y = α ( y ) , ˙ y = β y y + γ ( y ) , (92a)with α = − ( a + 4 b ) , β = − a , γ = 3 b , (92b)features the following explicit solution of its initial-values problem: y ( t ) = y (0)1 − α y (0) t , (93a) y ( t ) = n y (0) + (3 / 8) [ y (0)] o [1 − α y (0) t ] − β /α − (cid:18) (cid:19) (cid:20) y (0)1 − α y (0) t (cid:21) . (93b)The corresponding solution of the initial-values problem for the system of ODEs(72) is then obtained from this solution (93) via the relations y = − x − x , y = 3 x ( x + x ) (see (12)), which of course imply x ( t ) = − x ( t ) − y ( t ) , (94a)with x ( t ) given in terms of y ( t ) and y ( t ) (see (93)) as a solution of the( explicitly solvable !) second-degree equation6 ( x ) + 3 y x + y = 0 . (cid:4) (94b) ˙ y ˜ m = α + α y ˜ m , ˙ y ˜ m = β ( y ˜ m ) − m / ˜ m ) + β ( y ˜ m ) − m / ˜ m ) . (95a)Since we are mainly interested in the case when the right-hand sides of boththese ODEs are polynomial, we shall limit our consideration of this case to the3 subcases with ˜ m = 1 and to the single case with ˜ m = 2 , ˜ m = 4 . y ˜ m ( t ) the second-orderODE ¨ y ˜ m = α h β ( y ˜ m ) − m / ˜ m ) + β ( y ˜ m ) − m / ˜ m ) i , (95b)which is an autonomous Newtonian equation of motion (”acceleration equalforce”) and is of course solvable by quadratures (as discussed in more detail inthe following special cases).And of course once y ˜ m ( t ) is known, ˙ y ˜ m = [ ˙ y ˜ m ( t ) − α ] /α is as wellknown (see the first of the 2 ODEs (95a)). In this case ˜ m = 1 , ˜ m = 2 , or ˜ m = 2 , ˜ m = 4 , so that (95b) reads¨ w = α (cid:0) β w + β w (cid:1) ; (96)here and below w ( t ) stands for y ˜ m ( t ) respectively y ˜ m ( t ) , in the 2 cases ˜ m =1 , ˜ m = 2 , respectively ˜ m = 2 , ˜ m = 4.It is easily seen that the solution of the initial-value problem of this ODEreads as follows: w ( t ) = µ sn ( λ t + ρ, k ) , (97a)with the 4 parameters λ, µ, ρ and k determined by the following 4 formulasin terms of the 3 parameters α , β , β and the initial data w (0) = y (0), u (0) = y (0) respectively w (0) = y (0), u (0) = y (0) (in the 2 cases ˜ m = 1 , ˜ m = 2 , respectively ˜ m = 2 , ˜ m = 4): λ = − α β β k , µ = − k β β (1 + k ) , sn ( ρ, k ) = w (0) µ ,c k + c (cid:0) k (cid:1) = 0 , c = − α ( β ) ,c = ( α ) [ β + β u (0)] − α β β [ w (0)] − α ( β ) w (0)] . (97b)And of course correspondingly (see the first of the 2 ODEs (95a)) u ( t ) = ( α β ) − [ λµ cn ( λt + ρ, k ) dn ( λt + ρ, k ) − α β ] , (98)where of course u ( t ) stands for y ( t ) respectively y ( t ) . Here sn( z, k ) , cn( z, k ) , dn( z, k ) are the 3 standard Jacobian elliptic func-tions. In this case ˜ m = 1 , ˜ m = 3 , so that (95b) reads¨ y = α h β ( y ) + β ( y ) i , (99a)33mplying y ( t ) Z y (0) dy p C + ( α / y (2 β + β y ) = t ,C = [ y (0)] − ( α / 3) [ y (0)] n β + β [ y (0)] o . (99b)It is thus seen that in this case y ( t ) is a hyperelliptic function. In this case ˜ m = 1 , ˜ m = 4 , so that (95b) reads¨ y = α h β ( y ) + β ( y ) i , (100a)implying y ( t ) Z y (0) dy p C + ( α / y (2 β + β y ) = t ,C = [ y (0)] − ( α / 4) [ y (0)] n β + β [ y (0)] o . (100b)It is thus again seen that in this case y ( t ) is a hyperelliptic function. In this Appendix B we display—for the case M = 4 and N = 2—the ex-pressions of the time-derivatives ˙ y ˜ m ( t ) of the coefficients y ˜ m ( t ) in terms of thetime-derivatives ˙ y m ( t ) of the coefficients y m ( t ) and of the zeros x n ( t ), for the2 cases with µ = 3 , µ = 1 respectively µ = µ = 2 (for this terminology, see Section 2 ).In case (i), with µ = 3 , µ = 1 , these relations read as follows:˙ y = − x ˙ y + ˙ y x ) , ˙ y = − ( x ) ˙ y − ˙ y x ) , ˙ y = x ˙ y + 2 ˙ y ( x ) , (101a)˙ y = − x ) ˙ y + ˙ y x , ˙ y = − x ) ˙ y − ˙ y ( x ) , ˙ y = − x ˙ y + 3 ˙ y ( x ) , (101b)˙ y = − x ) ˙ y − x ˙ y , ˙ y = ( x ) ˙ y − y x , ˙ y = − ( x ) ˙ y + 3 ˙ y x , (101c)34 y = 2 ( x ) ˙ y + ( x ) ˙ y , ˙ y = 12 h ( x ) ˙ y − x ˙ y i , ˙ y = − h ( x ) ˙ y + 2 x ˙ y i . (101d)In case (ii), with µ = µ = 2 , these relations read as follows:˙ y = − ( x + x ) ˙ y + ˙ y ( x ) + x x + ( x ) , ˙ y = − x x ˙ y − ˙ y x x ( x + x ) , ˙ y = x x ˙ y + ( x + x ) ˙ y ( x x ) , (102a)˙ y = − h ( x ) + x x + ( x ) i ˙ y + ˙ y x + x , ˙ y = − x x ( x + x ) ˙ y − ˙ y x x , ˙ y = − x x ( x + x ) ˙ y + h ( x ) + x x + ( x ) i ˙ y ( x x ) , (102b)˙ y = − h ( x ) + x x + ( x ) i ˙ y − ( x + x ) ˙ y , ˙ y = ( x x ) ˙ y − ( x + x ) ˙ y x x , ˙ y = − ( x x ) ˙ y + h ( x ) + x x + ( x ) i ˙ y x x ( x + x ) , (102c)˙ y = x x ( x + x ) ˙ y + x x ˙ y , ˙ y = ( x x ) ˙ y − x x ˙ y x + x , ˙ y = − ( x x ) ˙ y + x x ( x + x ) ˙ y ( x ) + x x + ( x ) . (102d) References [1] F. Calogero, ”Motion of Poles and Zeros of Special Solutions of Nonlinearand Linear Partial Differential Equations, and Related ”Solvable” ManyBody Problems”, Nuovo Cimento , 177-241 (1978).[2] F. Calogero, Isochronous systems , Oxford University Press, Oxford, U. K.,(2008); paperback (2012).[3] D. 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