A Quasi-Lie Schemes Approach to Second-Order Gambier Equations

A quasi-Lie scheme is a geometric structure that provides t-dependent changes of variables transforming members of an associated family of systems of first-order differential equations into members of the same family. In this note we introduce two quasi-Lie schemes for studying second-order Gambier...

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Published in:	Symmetry, Integrability and Geometry: Methods and Applications
Date:	2013
Main Authors:	Cariñena, J.F., Guha, P., de Lucas, J.
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Language:	English
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spelling	Cariñena, J.F. Guha, P. de Lucas, J. 2019-02-19T19:03:08Z 2019-02-19T19:03:08Z 2013 A Quasi-Lie Schemes Approach to Second-Order Gambier Equations / J.F. Cariñena, P. Guha, L. de Lucas // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 56 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 34A26; 34A05; 34A34; 17B66; 53Z05 DOI: http://dx.doi.org/10.3842/SIGMA.2013.026 https://nasplib.isofts.kiev.ua/handle/123456789/149230 A quasi-Lie scheme is a geometric structure that provides t-dependent changes of variables transforming members of an associated family of systems of first-order differential equations into members of the same family. In this note we introduce two quasi-Lie schemes for studying second-order Gambier equations in a geometric way. This allows us to study the transformation of these equations into simpler canonical forms, which solves a gap in the previous literature, and other relevant differential equations, which leads to derive new constants of motion for families of second-order Gambier equations. Additionally, we describe general solutions of certain second-order Gambier equations in terms of particular solutions of Riccati equations, linear systems, and t-dependent frequency harmonic oscillators. This paper is a contribution to the Special Issue “Symmetries of Dif ferential Equations: Frames, Invariants and Applications”. The full collection is available at http://www.emis.de/journals/SIGMA/SDE2012.html. The research of J.F. Cari˜nena and J. de Lucas was supported by the Polish National Science Centre under the grant HARMONIA Nr 2012/04/M/ST1/00523. They also acknowledge partial financial support by research projects MTM–2009–11154 (MEC) and E24/1 (DGA). J. de Lucas would like to thank for a research grant FMI40/10 (DGA) to accomplish a research stay in the University of Zaragoza. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications A Quasi-Lie Schemes Approach to Second-Order Gambier Equations Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
title	A Quasi-Lie Schemes Approach to Second-Order Gambier Equations
spellingShingle	A Quasi-Lie Schemes Approach to Second-Order Gambier Equations Cariñena, J.F. Guha, P. de Lucas, J.
title_short	A Quasi-Lie Schemes Approach to Second-Order Gambier Equations
title_full	A Quasi-Lie Schemes Approach to Second-Order Gambier Equations
title_fullStr	A Quasi-Lie Schemes Approach to Second-Order Gambier Equations
title_full_unstemmed	A Quasi-Lie Schemes Approach to Second-Order Gambier Equations
title_sort	quasi-lie schemes approach to second-order gambier equations
author	Cariñena, J.F. Guha, P. de Lucas, J.
author_facet	Cariñena, J.F. Guha, P. de Lucas, J.
publishDate	2013
language	English
container_title	Symmetry, Integrability and Geometry: Methods and Applications
publisher	Інститут математики НАН України
format	Article
description	A quasi-Lie scheme is a geometric structure that provides t-dependent changes of variables transforming members of an associated family of systems of first-order differential equations into members of the same family. In this note we introduce two quasi-Lie schemes for studying second-order Gambier equations in a geometric way. This allows us to study the transformation of these equations into simpler canonical forms, which solves a gap in the previous literature, and other relevant differential equations, which leads to derive new constants of motion for families of second-order Gambier equations. Additionally, we describe general solutions of certain second-order Gambier equations in terms of particular solutions of Riccati equations, linear systems, and t-dependent frequency harmonic oscillators.
issn	1815-0659
url	https://nasplib.isofts.kiev.ua/handle/123456789/149230
citation_txt	A Quasi-Lie Schemes Approach to Second-Order Gambier Equations / J.F. Cariñena, P. Guha, L. de Lucas // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 56 назв. — англ.
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first_indexed	2025-11-24T15:04:59Z
last_indexed	2025-11-24T15:04:59Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 9 (2013), 026, 23 pages A Quasi-Lie Schemes Approach to Second-Order Gambier Equations? José F. CARIÑENA †, Partha GUHA ‡ and Javier DE LUCAS § † Department of Theoretical Physics and IUMA, University of Zaragoza, Pedro Cerbuna 12, 50.009, Zaragoza, Spain E-mail: jfc@unizar.es ‡ S.N. Bose National Centre for Basic Sciences, JD Block, Sector III, Salt Lake, Kolkata - 700.098, India E-mail: partha@bose.res.in § Faculty of Mathematics and Natural Sciences, Cardinal Stefan Wyszyński University, Wóy-cickiego 1/3, 01-938, Warsaw, Poland E-mail: j.delucasaraujo@uksw.edu.pl Received September 26, 2012, in final form March 14, 2013; Published online March 26, 2013 http://dx.doi.org/10.3842/SIGMA.2013.026 Abstract. A quasi-Lie scheme is a geometric structure that provides t-dependent changes of variables transforming members of an associated family of systems of first-order differential equations into members of the same family. In this note we introduce two quasi-Lie schemes for studying second-order Gambier equations in a geometric way. This allows us to study the transformation of these equations into simpler canonical forms, which solves a gap in the previous literature, and other relevant differential equations, which leads to derive new con- stants of motion for families of second-order Gambier equations. Additionally, we describe general solutions of certain second-order Gambier equations in terms of particular solutions of Riccati equations, linear systems, and t-dependent frequency harmonic oscillators. Key words: Lie system; Kummer–Schwarz equation; Milne–Pinney equation; quasi-Lie scheme; quasi-Lie system; second-order Gambier equation; second-order Riccati equation; superposition rule 2010 Mathematics Subject Classification: 34A26; 34A05; 34A34; 17B66; 53Z05 1 Introduction Apart from their inherent mathematical interest, differential equations are important due to their use in all branches of science [41, 50]. This strongly motivates their analysis as a means to study the problems they model. A remarkable approach to differential equations is given by geometric methods [47], which have resulted in powerful techniques such as Lax pairs, Lie symmetries, and others [48, 49]. A particular class of systems of ordinary differential equations that have been drawing some attention in recent years are the so-called Lie systems [1, 9, 33, 46, 54, 56]. Lie systems form a class of systems of first-order differential equations possessing a superposition rule, i.e. a func- tion that enables us to write the general solution of a first-order system of differential equations in terms of a generic collection of particular solutions and some constants to be related to initial conditions [7, 18]. The theory of Lie systems furnishes many geometric methods for studying these systems [6, 17, 19, 21, 22, 26, 27, 56]. For instance, superposition rules can be employed to simplify the use ?This paper is a contribution to the Special Issue “Symmetries of Differential Equations: Frames, Invariants and Applications”. The full collection is available at http://www.emis.de/journals/SIGMA/SDE2012.html mailto:jfc@unizar.es mailto:partha@bose.res.in mailto:j.delucasaraujo@uksw.edu.pl http://dx.doi.org/10.3842/SIGMA.2013.026 http://www.emis.de/journals/SIGMA/SDE2012.html 2 J.F. Cariñena, P. Guha and J. de Lucas of numerical techniques for solving differential equations [56], and the theory of reduction of Lie systems reduces the integration of Lie systems on Lie groups to solving Lie systems on simple Lie groups [6, 19]. The classification of systems admitting a superposition rule is due to Lie. His result, the nowadays called Lie–Scheffers theorem, states that a system admits a superposition rule if and only if it describes the integral curves of a t-dependent vector field taking values in a finite- dimensional Lie algebra of vector fields. The existence of such Lie algebras on R and R2 was analysed by Lie in his famous work [45]. More recently, the topic was revisited by Olver and coworkers [28], who clarified a number of details that were not properly described in the previous literature. Despite their interesting properties, Lie systems have a relevant drawback: there exist just a few Lie systems of broad interest [36]. Indeed, the Lie–Scheffers theorem and, more specifically, the classification of finite-dimensional Lie algebras of vector fields on low dimensional mani- folds [28, 45] clearly show that that being a Lie system is the exception rather than the rule. This has led to generalise the theory of Lie systems so as to tackle a larger family of remark- able systems [2, 3, 9, 36]. In particular, we henceforth focus on the so-called quasi-Lie schemes. These recently devised structures [15, 20] have been found quite successful in investigating trans- formation and integrability properties of differential equations, e.g. Abel equations, dissipative Milne–Pinney equations, second-order Riccati equations, and others [9]. In addition, the ob- tained results are useful so as to research on the physical and mathematical problems described through these equations. In this work, we study the second-order Gambier equations by means of the theory of quasi- Lie schemes. We provide two new quasi-Lie schemes. Their associated groups [15] give rise to groups of t-dependent changes of variables, which are used to transform second-order Gambier equations into another ones. Such groups allow us to explain in a geometric way the existence of certain transformations reducing a quite general subclass of second-order Gambier equations into simpler ones. Our approach provides a better understanding of a result pointed out in [30]. As a byproduct, we show that the procedure given in the latter work does not apply to every second-order Gambier equation, which solves a gap performed in there. We provide conditions for second-order Gambier equations, written as first-order systems, to be mapped into Lie systems via t-dependent changes of variables induced by our quasi-Lie schemes. This is employed to determine families of Gambier equations which can be transformed into second-order Riccati equations [31], Kummer–Schwarz equations [4, 5] and Milne–Pinney equations [25]. These results are employed to derive, as far as we know, new constants of motion for certain second-order Gambier equations. Moreover, the description of their general solutions in terms of particular solutions of t-dependent frequency harmonic oscillators, linear systems, or Riccati equations is provided [9, 29]. The structure of our paper goes as follows. Section 2 addresses the description of the fun- damental notions to be employed throughout our work. Section 3 describes a new quasi-Lie scheme for studying second-order Gambier equations. In Section 4 this quasi-Lie scheme is used to analyse the reduction of second-order Gambier equations to a simpler canonical form [30]. By using the theory of quasi-Lie systems, we determine in Section 5 a family of second-order Gambier equations that can be mapped into second-order Kummer–Schwarz equations. The investigation of constants of motion for some members of the previous family is performed in Section 6. In Section 7 we describe the general solutions of a family of second-order Gambier equations in terms of particular solutions of other differential equations. We present a second quasi-Lie scheme for investigating second-order Gambier equations in Section 8, and conditions are given to be able to transform these equations into second-order Riccati equations. Those second-order Gambier equations that can be transformed into second-order Riccati equations are integrated in Section 9. Finally, Section 10 is devoted to summarising our main results. A Quasi-Lie Schemes Approach to Second-Order Gambier Equations 3 2 Fundamentals Let us survey the fundamental results to be used throughout the work (see [15, 16, 17, 18] for details). In general, we hereafter assume all objects to be smooth, real, and globally defined on linear spaces. This simplifies our exposition and allows us to avoid tackling minor details. Given the projection π : (t, x) ∈ R × Rn 7→ x ∈ Rn and the tangent bundle projection τ : TRn → Rn, a t-dependent vector field on Rn is a mapping X : R × Rn → TRn such that τ ◦X = π. This condition entails that every t-dependent vector field X gives rise to a family {Xt}t∈R of vector fields Xt : x ∈ Rn 7→ X(t, x) ∈ TRn and vice versa. We call minimal Lie algebra ofX the smallest real Lie algebra V X containing the vector fields {Xt}t∈R. Given a finite- dimensional R-linear space V of vector fields on Rn, we write V (C∞(R)) for the C∞(R)-module of t-dependent vector fields taking values in V . An integral curve of X is a standard integral curve γ : R → R × Rn of its suspension, i.e. the vector field X = ∂/∂t+X(t, x) on R× Rn. Note that the integral curves of X of the form γ : t ∈ R→ (t, x(t)) ∈ R× Rn are the solutions of the system dxi dt = Xi(t, x), i = 1, . . . , n, (2.1) the referred to as associated system of X. Conversely, given such a system, we can define a t-dependent vector field on Rn [16] X(t, x) = n∑ i=1 Xi(t, x) ∂ ∂xi whose integral curves of the form (t, x(t)) are the solutions to (2.1). This justifies to write X for both a t-dependent vector field and its associated system. We call generalised flow a map g : (t, x) ∈ R × Rn 7→ gt(x) ∈ Rn such that g0 = IdRn . Every t-dependent vector field X can be associated with a generalised flow g satisfying that the general solution of X can be written in the form x(t) = gt(x0) with x0 ∈ Rn. Conversely, every generalised flow defines a vector field by means of the expression [15] X(t, x) = d ds ∣∣∣∣ s=t gs ◦ g−1t (x). Generalised flows act on t-dependent vector fields [18]. More precisely, given a generalised flow g and a t-dependent vector field X, we can define a unique t-dependent vector field, gFX, whose associated system has general solution x̄(t) = gt(x(t)), where x(t) is the general solution of X. In other words, every g induces a t-dependent change of variables x̄(t) = gt(x(t)) transforming the system X into gFX. Indeed, g can be viewed as a diffeomorphism ḡ : (t, x) ∈ R × Rn 7→ (t, gt(x)) ∈ R × Rn, and it can easily be proved that gFX is the only t-dependent vector field such that gFX = ḡ∗X, where ḡ∗ is the standard action of the diffeomorphism ḡ on vector fields (see [18]). Among all t-dependent vector fields, we henceforth focus on those whose associated systems are Lie systems. The characteristic property of Lie systems is to possess a superposition rule [7, 18, 46]. A superposition rule for a system X on Rn is a map Φ : (u(1), . . . , u(m); k1, . . . , kn) ∈ (Rn)m×Rn 7→ Φ(u(1), . . . , u(m); k1, . . . , kn) ∈ Rn allowing us to write its general solution x(t) as x(t) = Φ(x(1)(t), . . . , x(m)(t); k1, . . . , kn), for a generic family of particular solutions x(1)(t), . . . , x(m)(t) and a set of constants k1, . . . , kn to be related to initial conditions. 4 J.F. Cariñena, P. Guha and J. de Lucas The celebrated Lie–Scheffers theorem [46, Theorem 44] states that a system X possesses a superposition rule if and only if it is a t-dependent vector field taking values in a finite- dimensional real Lie algebra of vector fields, termed Vessiot–Guldberg Lie algebra [34, 53]. In other words, X is a Lie system if and only if V X is finite-dimensional [9]. This is indeed the main reason to define V X [14]. To illustrate the above notions, let us consider the Riccati equation [35] dx dt = b1(t) + b2(t)x+ b3(t)x 2, (2.2) where b1(t), b2(t), b3(t) are arbitrary functions of time. Its general solution, x(t), can be obtained from an expression [38, 56] x(t) = Φ(x(1)(t), x(2)(t), x(3)(t); k), where k is a real number to be related to the initial conditions of every particular solution, x(1)(t), x(2)(t), x(3)(t) are three different particular solutions of (2.2) and Φ : R3 × R → R is given by Φ(u(1), u(2), u(3); k) = u(1)(u(2) − u(3))− ku(2)(u(3) − u(1)) (u(2) − u(3))− k(u(3) − u(1)) . That is, the Riccati equations admit a superposition rule. Therefore, from the Lie–Scheffers Theorem, we infer that the t-dependent vector field X associated to a Riccati equation is such that V X is finite-dimensional. Indeed, X = ( b1(t) + b2(t)x+ b3(t)x 2 ) ∂ ∂x . Taking into account that X1 = ∂/∂x, X2 = x∂/∂x, X3 = x2∂/∂x span a finite-dimensional real Lie algebra V of vector fields and Xt = b1(t)X1+b2(t)X2+b3(t)X3, we obtain that {Xt}t∈R ⊂ V and V X becomes a (finite-dimensional) Lie subalgebra of V . The Lie–Scheffers theorem shows that just some few first-order systems are Lie sys- tems [9, 36]. For instance, this theorem implies that all Lie systems on the real line are, up to a change of variables, a particular case of a linear or Riccati equation [55]. Therefore, many other important differential equations cannot be studied through Lie systems (see [9] for exam- ples of this). In order to treat non-Lie systems, new techniques generalising Lie systems need to be developed. We here focus on the theory of quasi-Lie schemes [15, 20]. Definition 2.1. Let W , V be finite-dimensional real vector spaces of vector fields on Rn. We say that they form a quasi-Lie scheme S(W,V ) if: • W is a vector subspace of V . • W is a Lie algebra of vector fields, i.e. [W,W ] ⊂W . • W normalises V , i.e. [W,V ] ⊂ V . Associated to each quasi-Lie scheme, we have the C∞(R)-modules W (C∞(R)) and V (C∞(R)) of t-dependent vector fields taking values in W and V , respectively. Now, from the Lie alge- bra W , we define the group G(W ) of generalised flows of t-dependent vector fields taking values in W , the so-called group of the scheme. The relevance of this group is due to the following theorem [15]. Theorem 2.1. Given a quasi-Lie scheme S(W,V ), every generalised flow of G(W ) acts trans- forming elements of V (C∞(R)) into members of V (C∞(R)). A Quasi-Lie Schemes Approach to Second-Order Gambier Equations 5 In other words, the elements of the group of a scheme provide t-dependent changes of variables that transform systems of V (C∞(R)) into systems of this family. Roughly speaking, we can un- derstand this group as a generalisation of the t-independent symmetry group of a system: apart from transforming the initial system into itself, the transformations of the group of a scheme also may transform the initial system into one of the “same type”. For instance, given a Lie system X associated with a Vessiot–Guldberg Lie algebra V , then S(V, V ) becomes a quasi-Lie scheme. The group G(V ) allows us to transform X into a Lie system with a Vessiot–Guldberg Lie algebra V . This can be employed, for example, to transform Riccati equations into Riccati equations that can be easily integrated, giving rise to methods to integrate Riccati equations. In order to illustrate previous notions, we now turn to proving that quasi-Lie schemes allow us to cope with Abel equations of first-order and first kind [11], i.e. dx dt = b1(t) + b2(t)x+ b3(t)x 2 + b4(t)x 3, (2.3) with b1(t), . . . , b4(t) being arbitrary t-dependent functions. Indeed, if we fix W = 〈∂/∂x, x∂/∂x〉, it can be proved that S(W,V ) and V = 〈∂/∂x, x∂/∂x, x2∂/∂x, x3∂/∂x〉 is a quasi-Lie scheme and X ∈ V (C∞(R)) for every X related to an Abel equation (2.3). The elements of G(W ) transform Abel equations into Abel equations and geometrically recover the usual t-dependent changes of variables used to study these equations. This was employed in [11] to describe integrability properties of Abel equations. Given a quasi-Lie scheme S(W,V ), certain systems in V (C∞(R)) can be mapped into Lie systems admitting a Vessiot–Guldberg Lie algebra contained in V . This enables us to study the transformed system through techniques from the theory of Lie systems and, undoing the performed transformation, to obtain properties of the initial system under study [15]. Definition 2.2. Let S(W,V ) be a quasi-Lie scheme and X a t-dependent vector field in V (C∞(R)), we say that X is a quasi-Lie system with respect to S(W,V ) if there exists a gene- ralised flow g ∈ G(W ) and a Lie algebra of vector fields V0 ⊂ V such that gFX ∈ V0(C∞(R)). 3 A new quasi-Lie scheme for investigating second-order Gambier equations The Gambier equation [30, 32] can be described as the coupling of two Riccati equations in cascade, which can be given in the following form dy dt = −y2 + a1y + a2, dx dt = a0x 2 + nyx+ σ, where n is an integer, σ is a constant, which can be scaled to 1 unless it happens to be 0, and a0, a1, a2 are certain functions depending on time. The precise form of the coefficients of the Gambier equation is determined by singularity analysis, which leads to some constraints on a0, a1 and a2 [30]. For simplicity, we hereafter assume a0(0) 6= 0. Nevertheless, all our results can easily be generalised for the case a0(0) = 0. If n 6= 0, we can eliminate y between the two equations above, which gives rise to the referred to as second-order Gambier equation [32, 39, 40, 44], i.e. d2x dt2 = n− 1 xn ( dx dt )2 + a0 (n+ 2) n x dx dt + a1 dx dt − σ (n− 2) nx dx dt − a20 n x3 + ( da0 dt − a0a1 ) x2 + ( a2n− 2a0 σ n ) x− a1σ − σ2 nx . (3.1) 6 J.F. Cariñena, P. Guha and J. de Lucas The importance of second-order Gambier equations is due to their relations to remarkable differential equations such as second-order Riccati equations [31, 35], second-order Kummer– Schwarz equations [5, 16] and Milne–Pinney equations [32]. Additionally, by making appropriate limits in their coefficients, Gambier equations describe all the linearisable equations of the Painlevé–Gambier list [32]. Several particular cases of these equations have also been studied in order to analyse discrete systems [40]. Particular instances of (3.1) have already been investigated through the theory of Lie systems and quasi-Lie schemes. For instance, by fixing n = −2, σ = a1 = 0, and a0 to be a constant, the second-order Gambier equation (3.1) becomes a second-order Kummer–Schwarz equation (KS2 equation) [5, 16] d2x dt2 = 3 2x ( dx dt )2 − 2c0x 3 + 2ω(t)x, (3.2) where we have written c0 = −a20/4, with c0 a non-positive constant, and ω(t) = −a2(t) so as to keep, for simplicity in following procedures, the same notion as used in the literature, e.g. in [16]. The interest of KS2 equations is due to their relations to other differential equations of physical and mathematical interest [16, 24, 32]. For instance, for x > 0 the change of variables y = 1/ √ x transforms KS2 equations into Milne–Pinney equations, which frequently occur in cosmology [32]. Meanwhile, the non-local transformation dy/dt = xmaps KS2 equations into a particular type of third-order Kummer–Schwarz equations, which are closely related to Schwarzian derivatives [4, 16, 43]. Additionally, KS2 equations can be related, through the addition of the new variable dx/dt = v, to a Lie system associated to a Vessiot–Guldberg Lie algebra isomorphic to sl(2,R), which gave rise to various methods to study its properties and related problems [16]. If we now assume n = 1 and σ = 0 in (3.1), it results d2x dt2 = (a1 + 3a0x) dx dt − a20x3 + ( da0 dt − a0a1 ) x2 + a2x, which is a particular case of second-order Riccati equations [23, 31] that has been treated through the theory of quasi-Lie schemes and Lie systems in several works [10, 13, 29]. Furthermore, equations of this type have been broadly investigated because of its appearance in the study of the Bäcklund transformations for PDEs, their relation to physical problems, and the interest of the algebraic structure of their Lie symmetries [8, 23, 31, 37, 42]. In view of the previous results, it is natural to wonder which kind of second-order Gambier equations can be studied through the theory of quasi-Lie schemes. To this end, let us build up a quasi-Lie scheme for analysing these equations. As usual, the introduction of the new variable v ≡ dx/dt enables us to relate the second-order Gambier equation (3.1) to the first-order system dx dt = v, dv dt = (n− 1) n v2 x + a0 (n+ 2) n xv + a1v − σ (n− 2) n v x − a20 n x3 + ( da0 dt − a0a1 ) x2 + ( a2n− 2a0 σ n ) x− a1σ − σ2 nx , which is associated to the t-dependent vector field on TR0, with R0 ≡ R \ {0}, given by X = v ∂ ∂x + [ (n− 1) n v2 x + a0 (n+ 2) n xv + a1v − σ (n− 2) n v x − a20 n x3 A Quasi-Lie Schemes Approach to Second-Order Gambier Equations 7 + ( da0 dt − a0a1 ) x2 + ( a2n− 2a0 σ n ) x− a1σ − σ2 nx ] ∂ ∂v , termed henceforth the Gambier vector field. To obtain a quasi-Lie scheme for studying the above equations, we need to find a finite-dimensional R-linear space VG such that X ∈ VG(C∞(R)) for all a0, a1, a2, σ and n. Observe that X can be cast in the form X = 10∑ α=1 bα ( a0, da0 dt , a1, a2, σ, n ) Yα, (3.3) with b1, . . . , b10 being certain t-dependent functions whose form depends on the functions a0, da0/dt, a1, a2 and the constants σ and n. More specifically, these functions read b1 = 1, b2 = n− 1 n , b3 = a0 n+ 2 n , b4 = a1, b5 = −σn− 2 n , b6 = −a 2 0 n , b7 = da0 dt − a0a1, b8 = a2n− 2a0 σ n , b9 = −a1σ, b10 = −σ 2 n and Y1 = v ∂ ∂x , Y2 = v2 x ∂ ∂v , Y3 = xv ∂ ∂v , Y4 = v ∂ ∂v , Y5 = v x ∂ ∂v , Y6 = x3 ∂ ∂v , Y7 = x2 ∂ ∂v , Y8 = x ∂ ∂v , Y9 = ∂ ∂v , Y10 = 1 x ∂ ∂v . For convenience, we further define the vector field Y11 = x ∂ ∂x , which, although does not appear in the decomposition (3.3), will shortly become useful so as to describe the properties of Gambier vector fields. In view of (3.3), it easily follows that we can choose VG to be the space spanned by Y1, . . . , Y11. It is interesting to note that the linear space VG is not a Lie algebra as [Y3, Y6] does not belong to VG. Moreover, as adjY3Y6 ≡ j-times︷︸︸︷ [Y3, [Y3, [. . . , [Y3, Y6] . . .]]] = (−1)jxj+3 ∂ ∂v , j ∈ N, there is no finite-dimensional real Lie algebra V̂ ⊃ VG such that X ∈ V̂ (C∞(R)). Hence, X is not in general a Lie system, which suggests us to use quasi-Lie schemes to investigate it. To determine a quasi-Lie scheme involving VG, we must find a real finite-dimensional Lie algebra WG ⊂ VG such that [WG, VG] ⊂ VG. In view of Table 1, we can do so by setting WG = 〈Y4, Y8, Y11〉, which is a solvable three-dimensional Lie algebra. In fact, [Y4, Y8] = −Y8, [Y4, Y11] = 0, [Y8, Y11] = −Y8. In other words, we have proved the following proposition providing a new quasi-Lie scheme to study Gambier vector fields and, as shown posteriorly, second-order Gambier equations. Proposition 3.1. The spaces VG = 〈Y1, . . . , Y11〉 and WG = 〈Y4, Y8, Y11〉 form a quasi-Lie scheme S(WG, VG) such that X ∈ VG(C∞(R)) for every Gambier vector field X. Recall that Y11 is not necessary so that every Gambier vector field takes values in VG. Hence, why it is convenient to add it to VG? One reason can be found in Table 1. If VG had not contained Y11, then S(WG, VG) would have not been a quasi-Lie scheme as WG * VG. In addition, Y8 could not belong to WG neither, as [Y8, Y1] ∈ VG provided Y11 − Y4 ∈ VG. Hence, including Y11 in VG allows us to choose a larger WG ⊂ VG. In turn this gives rise to a larger group G(WG), which will be of great use in following sections. 8 J.F. Cariñena, P. Guha and J. de Lucas Table 1. Lie brackets [Yi, Yj ] with i = 4, 8, 11 and j = 1, . . . , 11. [·, ·] Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9 Y10 Y11 Y4 Y1 Y2 0 0 0 −Y6 −Y7 −Y8 −Y9 −Y10 0 Y8 Y11 − Y4 2Y4 Y7 Y8 Y9 0 0 0 0 0 −Y8 Y11 −Y1 −Y2 Y3 0 −Y5 3Y6 2Y7 Y8 0 −Y10 0 4 Transformation properties of second-order Gambier equations Remind that Theorem 2.1 states that the t-dependent changes of variables associated to the elements of the group G(W ) of a quasi-Lie scheme S(W,V ) establish bijections among the t- dependent vector fields taking values in V . This may be of great use so as to transform their associated systems into simplified forms, e.g. in the case of Abel equations [11]. We next show how this can be done for studying the transformation of second-order Gambier equations into simpler ones whose corresponding coefficients a1 vanish [30]. This retrives known results from a geometrical viewpoint and shows that certain second-order Gambier equations cannot be transformed into simpler ones, solving a small gap performed in [30]. As the vector fields in WG span a finite-dimensional real Lie algebra of vector fields on TR0, there exists a local Lie group action ϕ : G × TR0 → TR0 whose fundamental vector fields are the elements of WG. By integrating the vector fields of WG (see [12] for details), the action can easily be written as ϕ ( g, ( x v )) = ( αx γx+ δv ) , where g ∈ Td ≡ {( α 0 γ δ ) ∣∣∣∣α, δ ∈ R+, γ ∈ R } . The theory of Lie systems [9, 17] states that the solutions of a system associated to a t-dependent vector field taking values in the real Lie algebraWG are of the form (x(t), v(t)) = ϕ(h(t), (x0, v0)), with h(t) being a curve in Td with h(0) = e. Therefore, every g ∈ G(WG) can be written as gt(·) = ϕ(h(t), ·) for a certain curve h(t) in G with h(0) = e. Conversely, given a curve h(t) in Td with h(0) = e, the curve (x(t), v(t)) = ϕ(h(t), (x0, v0)) is the general solution of a system of WG(C∞(R)), which leads to a generalised flow gt(·) = ϕ(h(t), ·) of G(WG) [9, 17]. Hence, the elements of G(WG) are generalised flows of the form gh(t)(t, x, v) = ϕ(h(t), x, v), for h(t) any curve in Td with h(0) = e. Observe that every h(t) is a matrix of the form h(t) = ( α(t) 0 γ(t) δ(t) ) where α(t), δ(t) and γ(t) are t-dependent functions such that α(0) = δ(0) = 1 and γ(0) = 0, because h(0) = e, and α(t) > 0, δ(t) > 0 as h(t) ∈ Td for every t ∈ R. Hence, every element of G(WG) is of the form gα(t),γ(t),δ(t)(t, x, v) ≡ gh(t)(t, x, v) = (t, α(t)x, γ(t)x+ δ(t)v). (4.1) Theorem 2.1 implies that for every g ∈ G(WG) and Gambier vector field X ∈ VG(C∞(R)), we have gFX ∈ VG(C∞(R)). More specifically, a long but straightforward calculation shows that gFX = 11∑ α=1 b̄α(t)Yα, (4.2) A Quasi-Lie Schemes Approach to Second-Order Gambier Equations 9 where the functions b̄α = b̄α(t), with α = 1, . . . , 11, are b̄1 = α δ , b̄2 = n− 1 n α δ , b̄3 = a0(n+ 2) nα , b̄4 = a1 + 1 δ dδ dt + (2− n)γ nδ , b̄5 = (2− n)σα n , b̄6 = − a 2 0δ nα3 , b̄7 = 1 α2 ( δ da0 dt − a0a1δ − n+ 2 n a0γ ) , b̄8 = δ α ( na2 − 2σ n a0 − γ δ a1 − γ2 nδ2 − γ δ2 dδ dt + 1 δ dγ dt ) , b̄9 = −σ ( a1δ + (2− n)γ n ) , b̄10 = −σ 2αδ n , b̄11 = 1 α dα dt − γ δ . (4.3) Let us use this so as to transform the initial Gambier vector field into another one that is related, up to a t-reparametrisation τ = τ(t), to a second-order Gambier equation with τ -dependent coefficients ā0, ā1, ā2, a constant σ̄ and an integer number n̄. Additionally, we impose ā1 = 0 in order to reduce our initial first-order Gambier equation into a simpler one. In this way, we have gFX = ξ(t) [ Y1 + n̄− 1 n̄ Y2 + ā0 n̄+ 2 n̄ Y3 − σ n̄− 2 n̄ Y5 − ā20 n̄ Y6 + dā0 dτ Y7 + ( ā2n̄− 2ā0 σ̄ n̄ ) Y8 − σ̄2 n̄ Y10 ] , (4.4) for a certain non-vanishing function ξ(t) = dτ/dt. Therefore, b̄4 = b̄9 = b̄11 = 0, i.e. a1 + 1 δ dδ dt + (2− n)γ nδ = 0, (4.5) σ ( a1δ + (2− n)γ n ) = 0, (4.6) 1 α dα dt − γ δ = 0. (4.7) As we want our method to work for all values of σ, e.g. σ 6= 0, equation (4.6) implies a1δ + (2− n)γ n = 0. (4.8) As δ > 0, the above equation involves that a1 = 0 for n = 2. In other words, we cannot transform a Gambier vector field with a1 6= 0 into a new one with ā1 = 0 through our methods if n = 2 and σ 6= 0. In view of this, let us assume that n 6= 2. From (4.5) and (4.8), and using again that δ > 0, we infer that dδ/dt = 0. As δ(0) = 1, then δ = 1. Plugging the value of δ into (4.7) and (4.8), we obtain 1 α dα dt = na1 n− 2 = γ ⇐⇒ α = exp (∫ t 0 na1 n− 2 dt′ ) , γ = na1 n− 2 , which fixes the form of g mapping a system X into (4.4). Bearing previous results in mind, we see that the non-vanishing t-dependent coefficients (4.3) become b̄1 = α, b̄2 = n− 1 n α, b̄3 = a0(n+ 2) nα , b̄5 = (2− n)σα n , b̄6 = − a20 nα3 , b̄7 = d dt ( a0 α2 ) , b̄8 = 1 α ( na2 − 2σa0 n − n(n− 1) (n− 2)2 a21 + n n− 2 da1 dt ) , b̄10 = −σ 2α n . 10 J.F. Cariñena, P. Guha and J. de Lucas Comparing this and (4.4), we see that to transform the initial first-order Gambier equation into a new one through a t-dependent change of variables (4.1) and a t-reparametrisation dτ = αdt requires ξ = α. The resulting system reads dx̄ dτ = v̄, dv̄ dτ = (n− 1) n v̄2 x̄ + a0 (n+ 2) α2n x̄v̄ − σ (n− 2) n v̄ x̄ − a20 α4n x̄3 + d ( a0/α 2 ) dτ x̄2 + ( ā2n− 2a0α −2σ n ) x̄− σ2 nx̄ , where ā2 = 1 α2 ( a2 − (n− 1) (n− 2)2 a21 + 1 n− 2 da1 dt ) . Therefore, redefining n̄ = n, ā0 = a0/α 2 and σ̄ = σ, we obtain that the above system is associated to a second-order Gambier equation with ā1 = 0. Meanwhile, as g induces a t- dependent change of variables given by x̄ = αx, v̄ = dα dτ x+ v, we see that this t-dependent change of variables can be viewed as a t-dependent change of variables x̄ = αx along with a t-reparametrisation t→ τ such that v̄ = dx̄/dτ . Indeed, x̄ = αx =⇒ dx̄ dτ = dα dτ x+ α dx dτ = dα dτ x+ v = v̄. Furthermore, these transformations map the initial second-order Gambier equation with n 6= 2 into a new one with ā1 = 0, i.e. depending only on two functions ā0 and ā2 and the constants σ and n. We can therefore formulate the following result. Proposition 4.1. Every second-order Gambier equation (3.1) with n 6= 2 can be transformed via a t-dependent change of variables x̄ = α(t)x and a t-reparametrisation τ = τ(t), with dτ = αdt, α(0) = 1 and 1 α dα dt = n n− 2 a1, into a second-order Gambier equation whose a1 vanishes and n, σ remain the same. In the above proposition, we excluded second-order Gambier equations with n = 2 as we noticed that in this case the proof of this proposition does not hold: we cannot transform an initial Gambier vector field with σ 6= 0 and a1 6= 0 into a new one with ā1 = 0. Moreover, it is easy to see that the transformation provided in Proposition 4.1 does not exist for n = 2 and a1 6= 0. This was not noticed in [30], where this transformation is wrongly claimed to transform any Gambier equation into a simpler one with a1 = 0. In view of this, we cannot neither ensure, as claimed in [30], that second-order Gambier equations are not given in their simplest form. 5 Quasi-Lie systems and Gambier equations The theory of quasi-Lie schemes mainly provides information about quasi-Lie systems, which can be mapped to Lie systems through one of the transformations of the group of a quasi-Lie scheme. This allows us to employ the techniques of the theory of Lie systems so as to study the A Quasi-Lie Schemes Approach to Second-Order Gambier Equations 11 obtained Lie systems, and, undoing the performed t-dependent change of variables, to describe properties of the initial system [15]. Motivated by the above, we determine and study the Gambier vector fields X ∈ V (C∞(R)) which are quasi-Lie systems relative to S(WG, VG). This task relies in finding triples (g,X, V0), with g ∈ G(WG) and V0 being a real Lie algebra included in VG in such a way that gFX ∈ V0(C ∞(R)). One of the key points to determine quasi-Lie systems is to find a Lie algebra V0. In the case of Gambier vector fields, this can readily be obtained by recalling that certain instances of second-order Gambier equations, e.g. (3.2), are particular cases of KS2 equations. By adding a new variable v ≡ dx/dt to (3.2), the resulting first-order system becomes a Lie system (see [16]) related to a three-dimensional Vessiot–Guldberg Lie algebra V0 ⊂ VG of vector fields on TR0 spanned by X1 = 2x ∂ ∂v , X2 = x ∂ ∂x + 2v ∂ ∂v , X3 = v ∂ ∂x + ( 3 2 v2 x − 2c0x 3 ) ∂ ∂v , (5.1) i.e. X1 = 2Y8, X2 = Y11 + 2Y4, X3 = Y1 + 3 2 Y2 − 2 c0 Y6. Consequently, it makes sense to look for Gambier vector fields X ∈ V (C∞(R)) that can be transformed, via an element g ∈ G(WG), into a Lie system gFX ∈ V0(C∞(R)), i.e. gFX = 2f(t)Y8 + g(t)(Y11 + 2Y4) + h(t) ( Y1 + 3 2 Y2 − 2c0Y6 ) , for certain t-dependent functions f , g and h. Comparing the expression of gFX given by (4.3) and the above, we find that gFX ∈ V0(C∞(R)) if and only if b̄3 = b̄5 = b̄7 = b̄9 = b̄10 = 0 (5.2) and b̄4 = 2b̄11, b̄2 = 3 2 b̄1, b̄6 = −2c0b̄1. (5.3) From expressions (4.3) and remembering that α > 0 and δ > 0, we see that condition b̄10 = 0 implies that σ = 0. This involves, along with (4.3), that b̄5 = b̄9 = 0. Meanwhile, from b̄2 = 3b̄1/2 we obtain n = −2, which in turn ensures that b̄3 = 0. Bearing all this in mind, b̄7 = 0 reads a0a1 − da0 dt = 0. (5.4) Above results impose restrictions on the form of the Gambier vector field X to be able to be transformed into a Lie system possessing a Vessiot–Guldberg Lie algebra V0 via an element g ∈ G(WG). Let us show that the remaining conditions in (5.2) and (5.3) merely characterise the form of the t-dependent change of variables g and the coefficient c0 appearing in (3.2). Conditions 2b̄11 = b̄4 and b̄6 = −2c0b̄1 read d dt log α2 δ = a1, a20 = −4c0 α4 δ2 . (5.5) Using the first condition above, the relation (5.4), and taking into account that α(0) = δ(0) = 1 and δ, α > 0, we see that d dt(α 2δ−1) α2δ−1 = 1 a0 da0 dt =⇒ α2 δ = a0 a0(0) . 12 J.F. Cariñena, P. Guha and J. de Lucas Using the second equality in (5.5), we obtain 4c0 = −a0(0)2, which fixes c0 in terms of a coefficient of the initial second-order Gambier equation. Concerning the t-dependent coefficients b̄1, . . . , b̄11, the non-vanishing ones under above con- ditions, i.e. b̄1, b̄2, b̄4, b̄6, b̄8 and b̄11, can readily be obtained through relations (5.3) and b̄1 = α δ , b̄8 = δ α ( −2a2 − γ δa0 da0 dt + γ2 2δ2 − γ δ2 dδ dt + 1 δ dγ dt ) , b̄11 = 1 α dα dt − γ δ . In other words, we have proved the following proposition. Proposition 5.1. A Gambier vector field X is a quasi-Lie system relative to S(WG, VG) that can be transformed into a Lie system taking values in a Lie algebra of the form V0 if and only if a0a1 = da0/dt, n = −2 and σ = 0. Under these conditions, the constant c0 appearing in X3 becomes c0 = −a0(0)2/4. Additionally, a transformation g(α(t), γ(t), δ(t)) ∈ G(WG) maps X into gFX ∈ V0(C∞(R)) if and only if α2/δ = a0/a0(0). More specifically, gFX reads gFX = α δ X3 + ( 1 α dα dt − γ δ ) X2 + δ 2α ( −2a2 − γ δa0 da0 dt + γ2 2δ2 − γ δ2 dδ dt + 1 δ dγ dt ) X1. (5.6) The above proposition allows us to determine the transformations g ∈ G(WG) ensuring that a Gambier vector field and its related second-order Gambier equation can be mapped, maybe up to a t-reparametrisation, into a new Gambier vector field related to a KS2 equation. Indeed, from (3.2), we see that to do so, we need to impose gFX = ξ(t)(X3 + ω(t)X1), for a certain function ω(t) and a function ξ(t) such that ξ(t) 6= 0 for all t ∈ R. Comparing (5.6) with the above expression, we see that 1 α dα dt − γ δ = 0, (5.7) which, in view of the fact that gα(t),γ(t),δ(t) satisfies α2 δ = a0 a0(0) , (5.8) enables us to determine the searched transformations. In fact, fixed a non-vanishing t-dependent function α with α(0) = 1, the above conditions determine the values of δ and γ of g. Now, a t-dependent reparametrisation τ = ∫ t 0 α δ dt′ (5.9) transforms the system associated to gFX into a new system related to X3 + ω(t(τ))X1, where ω is given by ω = − δ2 2α2 ( 2a2 + γ a0δ da0 dt − γ2 2δ2 − 1 δ dγ dt + γ δ2 dδ dt ) . (5.10) Proposition 5.2. Every Gambier vector field X satisfying a0a1 = da0/dt, n = −2 and σ = 0 is a quasi-Lie system relative to S(WG, VG) that can be transformed into a Lie system associa- ted to X3 + ω(t)X1, with c0 = −a0(0)2/4 and ω(t) given by (5.10), through a transformation gα(t),γ(t),δ(t) ∈ G(WG), whose coefficients are given by any solution of (5.7) and (5.8), and the t-dependent reparametrisation (5.9). A Quasi-Lie Schemes Approach to Second-Order Gambier Equations 13 Note that the transformation gα(t),γ(t),δ(t) can be viewed as a t-dependent change of variables x̄ = αx and a t-reparametrisation dτ = α/δ dt, i.e. x̄ = αx, v̄ = γx+ δv, and, in view of the first condition in (5.7), we see that dx̄ dτ = dα dτ x+ α dx dτ = γx+ δv = v̄. From this and Proposition 5.2, we obtain the following proposition about second-order Gambier equations. Corollary 5.1. Every second-order Gambier equation of the form d2x dt2 = 3 2x ( dx dt )2 + 1 a0 da0 dt dx dt + a20 2 x3 − 2a2x (5.11) can be mapped into a KS2 equation d2x̄ dτ2 = 3 2x̄ ( dx̄ dτ )2 + 1 2 a0(0)2x̄3 − δ2 α2 ( 2a2 + γ a0δ da0 dt − 1 2 γ2 δ2 − 1 δ dγ dt + γ δ2 dδ dt ) x̄ (5.12) through a t-dependent change of variables x̄(t) = α(t)x(t), where α(t) is any positive function with α(0) = 1, and a t-reparametrisation τ(t) with dτ = α/δdt, with δ and γ being determined from α by the relations δ = α2a0(0) a0 , γ = αa0(0) a0 dα dt . (5.13) 6 Constants of motion for second-order Gambier equations In this section, we obtain constants of motion for second-order Gambier equations. We do so by analysing the existence of t-independent constants of motion for systems gFX ∈ V0(C∞(R)), with X being a Gambier vector field. By undoing the t-dependent change of variables g, this leads to determining constants of motion for a Gambier vector field X and its corresponding second-order Gambier equation. Let F : TR0 → R be a t-independent constant of motion for gFX, we have (gFX)tF = 0 for all t ∈ R. This involves that F is a t-independent constant of motion for all successive Lie brackets of elements of {(gFX)t}t∈R as well as their linear combinations. In other words, F is a common first-integral for the vector fields V gFX ⊂ V0. When ω(t) is not a constant, it can be verified that V gFX = V0. Thus, if F is a first-integral for all these vector fields, it is so for all vector fields contained in the generalised distribution Dp = 〈(X1)p, (X2)p, (X3)p〉, with p ∈ TR0. Hence, dFp ∈ D◦p, i.e. dFp is incident to all vectors of Dp. In this case, Dp = TpTR0 for a generic p ∈ TR0, which implies that dFp = 0 at almost every point. Since F is assumed to be differentiable, we have that F is constant on each connected component of TR0 and gFX has no non-trivial t-independent constant of motion. If ω(t) = λ for a real constant λ, then dimDp = 1 at a generic point and it makes sense to look for non-trivial t-dependent constants of motion for gFX. In view of (5.10), this condition implies λ = −a0(0)2α2 2a20 [ 2a2 + 1 a0 da0 dt d logα dt − 1 2 ( d logα dt )2 − d2 logα dt2 ] . 14 J.F. Cariñena, P. Guha and J. de Lucas The function F can therefore be obtained by integrating v ∂F ∂x + ( 3 2 v2 x + a0(0)2 2 x3 + 2λx ) ∂F ∂v = 0. In employing the characteristics method [10], we find that F must be constant along the integral curves of the so-called characteristic system, namely dx v = dv 3 2 v2 x + a0(0)2 2 x3 + 2λx . Let us focus on the region with v 6= 0, i.e. O ≡ {(x, v) ∈ TR0 \| v 6= 0}. We obtain from the previous equations that dv dx = 3 2 v x + a0(0)2 2 x3 v + 2λ x v . Let us focus on the case x > 0; the other case can be obtained in a similar way and leads to the same result. Multiplying on right and left by v/x and defining w ≡ v2 and z ≡ x2, we obtain dw dz = 3 2 w z + a0(0)2 2 z + 2λ. As this equation is linear, its general solution can be easily derived to obtain w(z) = ( a0(0)2z1/2 − 4λz−1/2 + ξ ) z3/2, for an arbitrary real constant ξ. From here, it results −a0(0)2x+ v2 x3 + 4λ x = ξ. Consequently, F is any function of the form F = F (ξ), for instance, F = −a0(0)2x+ v2 x3 + 4λ x , (x, v) ∈ O. In principle, F was defined only on O. Nevertheless, as this region is an open and dense subset of TR0, and in view of the expression for F , we can extend it differentiably to TR0 in a unique way. Since F is a constant of motion on O, it trivially becomes so on the whole TR0. Summarising, we have proved the following. Theorem 6.1. A second-order Gambier equation d2x dt2 = 3 2x ( dx dt )2 + 1 a0 da0 dt dx dt + a20 2 x3 − 2a2x, (6.1) admits a constant of motion of the form F = −a0(0)2x̄+ v̄2 x̄3 + 4λ x̄ , (6.2) where λ is a real constant, x̄ = αx and v̄ = δdx/dt + γx, with α being a particular positive solution with α(0) = 1 of d2 logα dt2 = 2λa20 a0(0)2α2 + 2a2 + 1 a0 da0 dt d logα dt − 1 2 ( d logα dt )2 (6.3) and γ, δ are determined from α by means of the conditions (5.13). A Quasi-Lie Schemes Approach to Second-Order Gambier Equations 15 The above proposition can be employed to derive a constant of motion for certain families of second-order Gambier equations. For instance, if we start by a Gambier equation (6.1) with a2 = −λa20/a0(0)2, for a certain real constant λ, then α = 1 is a solution of (6.3). In view of (5.13), γ = 0 and δ = a0(0)/a0. Therefore, Theorem 6.1 establishes that the second-order Gambier equation (6.1) admits a constant of motion F = −a0(0)2x+ a0(0)2 a20x 3 ( dx dt )2 + 4λ x . (6.4) Consider now a general second-order Gambier equation (6.1) and let us search for a constant of motion (6.2) with λ = 0. In this case, (6.3) can be brought into a Riccati equation dw dt = 2a2 + 1 a0 da0 dt w − 1 2 w2, where w ≡ d logα/dt, whose solutions can be investigated through many methods [19, 22]. The derivation of a particular solution provides a constant of motion for the second-order Gambier equation (6.1) that can be obtained through the previous theorem. Additionally, this particular solution can be used to obtain the general solution of the Riccati equation [9, 19], which in turns could be used to derive new constants of motion for the second-order Gambier equation (6.1). Note that all the above procedure depends deeply on the fact that λ is a constant. Recall that if λ is not a constant, then gFX does not admit any t-independent constant of motion. Nevertheless, other methods can potentially be applied in this case. For instance, using that S(V0, V0) is a quasi-Lie scheme, we can derive the group G(V0) of this scheme and to use an element h ∈ G(V0) to transform gFX into other Lie system hFgFX of the same type, e.g. another of the form hFgFX = ξ2(t)(c1X1 + c2X2 + c3X3), with c1, c2, c3 ∈ R, the vector fields X1, X2, X3 are those of (5.1), and ξ2(t) is any t-dependent nonvanishing function. As this system is, up to a t-parametrization, a t-independent vector field, it admits a local t-independent constant of motion. By inverting the t-dependent changes of variables h and g, it gives rise to a t-dependent constant of motion of X and the corresponding second-order Gambier equation. 7 Second-order Gambier and Milne–Pinney equations Consider the KS2 equation (5.12) with x > 0 (we can proceed analogously for the case x < 0). The change of variables x = 1/y2, with y > 0, transforms it into a Milne–Pinney equation [52] d2y dτ2 = −ω(t(τ))y − a0(0)2 4y3 . (7.1) These equations admit a description in terms of a Lie system related to a Vessiot–Guldberg Lie algebra isomorphic to sl(2,R) [12, 56]. This was employed in several works to describe their general solutions in terms of particular solutions of the same or others differential equations, e.g. Riccati equations and t-dependent harmonic oscillators [9]. Previous results allow us to describe the general solution of (5.11) in terms of particular solutions of Riccati equations or t-dependent frequency harmonic oscillators. Indeed, in view of Corollary 5.1, these equations can be transformed into a KS2 equation through a t-dependent change of variables x̄(t) = α(t)x(t) and a t-reparametrisation dτ = (α/δ) dt. In turn, y = 1/ √ x̄ transforms (5.12) into (7.1), whose general solution y(t) can be written as [12] y(τ) = √ k1z21(τ) + k2z22(τ) + 2C(k1, k2,W )z1(τ)z2(τ), 16 J.F. Cariñena, P. Guha and J. de Lucas where C2(k1, k2,W ) = k1k2 +a0(0)2/(4W 2), the functions z1(τ), z2(τ) are two linearly indepen- dent solutions of the system d2z dτ2 = −ω(t(τ))z, and W is the Wronskian related to such solutions. Inverting previous changes of variables, the general solution for any second-order Gambier equation (6.1) reads x(t) = α−1 [ k1z 2 1(τ(t)) + k2z 2 2(τ(t))± 2 √ k1k2 + a0(0)2/(4W 2)z1(τ(t))z2(τ(t)) ]−1 . Therefore, we have proved the following proposition: Proposition 7.1. The general solution of a second-order Gambier equation d2x dt2 = 3 2x ( dx dt )2 + 1 a0 da0 dt dx dt + a20 2 x3 − 2a2x, (7.2) can be brought into the form x(t) = α−1 [ k1z 2 1(τ(t)) + k2z 2 2(τ(t))± 2 √ k1k2 + a0(0)2/(4W 2)z1(τ(t))z2(τ(t)) ]−1 , where z1(τ) and z2(τ) are particular solutions of the τ -dependent harmonic oscillator d2z dτ2 = −ω(t(τ))z = − δ2 2α2 ( 2a2 + γ a0δ da0 dt − 1 2 γ2 δ2 − 1 δ dγ dt + γ δ2 dδ dt ) z, and W = z1 dz2/dτ − z2 dz1/dτ , with dτ = α/δ dt and α, δ and γ certain t-dependent functions satisfying (5.13). Many other similar results can be obtained in an analogous manner. For instance, the theo- ry of Lie systems was also used in [12] to prove that the general solution of a Milne–Pinney equation (7.1) can be written as y(τ) = √ [k1(x1(τ)− x2(τ))− k2(x1(τ)− x3(τ))]2 − a0(0)2[x2(τ)− x3(τ)]/4 (k2 − k1)(x2(τ)− x3(τ))(x2(τ)− x1(τ))(x1(τ)− x3(τ)) , where x1(τ), x2(τ) and x3(τ) are three different particular solutions of the Riccati equation dx dτ = −ω(t(τ))− x2. Proceeding as before, we can describe the general solution of a second-order Gambier equa- tion (7.2) in terms of solutions of these Riccati equations. In addition, by applying the theory of Lie systems [19] to Milne–Pinney equations (7.1), we can obtain many other results about subfamilies of second-order Gambier equations. In addition, the relation between second-order Gambier equations and Milne–Pinney equations enables us to obtain several other results in a simple way. Proposition 7.2. The second-order Gambier equation (5.11) with 2a2 = a20 > 0, i.e. d2x dt2 = 3 2x ( dx dt )2 + 1 a0 da0 dt dx dt + a20 2 x3 − a20x, (7.3) can be transformed into the integrable Milne–Pinney equation d2y dτ2 − y 2 + 1 4y3 = 0 (7.4) under the transformation x = 1/y2 and a t-reparametrisation dτ = a0dt. A Quasi-Lie Schemes Approach to Second-Order Gambier Equations 17 Note 7.1. For second-order Gambier equations (5.11) with a0 = 0, the transformations given in the above proposition map these equations into harmonic oscillators. Proposition 7.3. A constant of motion of (7.4) is given by IMP = 1 2 [( dy dτ )2 − ( y2 2 + 1 4y2 )] . Using y = x−1/2 transformation, we obtain a constant of motion for the special second-order Gambier equation (7.3) of the form I2G = 1 4 1 x3a20 ( dx dt )2 − ( 1 2x + x 4 ) . (7.5) Thus, we say the special equation (7.3), which yields a constant of motion, is an integrable deformation of the Milne–Pinney equation. Note 7.2. Recall that Theorem 6.1 can be applied to those equations (5.11) where a2 = −λa20/a20(0) giving rise to the constant of motion (6.4). Observe that (7.3) is of this type with λ = −a0(0)2/2. Then, the constant of motion (6.4) is, up to a multiplicative constant, the same as (7.5). Despite this, the above illustrates how certain results can readily be obtained through Milne–Pinney equations. 8 A second quasi-Lie schemes approach to second-order Gambier equations Apart from our first approach, we can provide a second quasi-Lie scheme to study second-order Gambier equations. This one is motivated by the fact that some cases of these equations, namely d2x dt2 = − ( 3x dx dt + x3 ) + f(t) + g(t)x+ h(t) ( x2 + dx dt ) , for arbitrary t-dependent functions f , g and h, form a particular subclass of second-order Riccati equations that are SODE Lie systems [13, 16]. That is, when written as first-order systems dx dt = v, dv dt = − ( 3xv + x3 ) + f(t) + g(t)x+ h(t) ( x2 + v ) , (8.1) they become Lie systems. Indeed, these systems possess a Vessiot–Guldberg Lie algebra V0 isomorphic to sl(3,R) given by [10, 29] X1 = v ∂ ∂x − ( 3xv + x3 ) ∂ ∂v , X2 = ∂ ∂v , X3 = − ∂ ∂x + 3x ∂ ∂v , X4 = x ∂ ∂x − 2x2 ∂ ∂v , X5 = ( v + 2x2 ) ∂ ∂x − x ( v + 3x2 ) ∂ ∂v , X6 = 2x ( v + x2 ) ∂ ∂x + 2 ( v2 − x4 ) ∂ ∂v , X7 = ∂ ∂x − x ∂ ∂v , X8 = 2x ∂ ∂x + 4v ∂ ∂v . In terms of these vector fields, (8.1) is the associated system to the t-dependent vector field X1 + f(t)X2 + 1 2 g(t)(X3 +X7) + h(t) 4 (−2X4 +X8). 18 J.F. Cariñena, P. Guha and J. de Lucas Table 2. Lie brackets [Yi, Yj ] with i = 4, 8, 11 and j = 12, . . . , 17. [·, ·] Y12 Y13 Y14 Y15 Y16 Y17 Y4 0 0 Y14 0 −Y16 Y17 Y8 −Y9 −Y7 Y13 − Y3 −Y6 0 2Y3 Y11 −Y12 Y13 0 2Y15 4Y16 0 Therefore, it is natural to analyse which Gambier vector fields X can be transformed through a quasi-Lie scheme into one of these Lie systems. Despite the interest of the previous idea, the quasi-Lie scheme provided in the previous section cannot be employed to determine all quasi-Lie systems of this type. The reason is that V0 is not included in VG and, therefore, there exists no g ∈ G(WG) such that gFX ∈ V0(C∞(R)) ⊂ VG(C∞(R)). To solve this drawback, we now determine a new quasi-Lie scheme S(WG, V ′ G) such that the space V ′G contains VG + V0. This can be done by defining V ′G as the R-linear space generated by the elements of VG and the vector fields Y12 = ∂ ∂x , Y13 = x2 ∂ ∂x , Y14 = xv ∂ ∂x , Y15 = x3 ∂ ∂x , Y16 = x4 ∂ ∂v , Y17 = v2 ∂ ∂v . Indeed, using Tables 1 and 2, we readily obtain that [WG, V ′ G] ⊂ V ′G. Moreover, as we already know that WG is a Lie algebra, S(WG, V ′ G) becomes a quasi-Lie scheme. Since we impose that gFX, which is given by (4.2), must be of the form (8.1) up to a t- reparametrisation, we obtain gFX = 11∑ α=1 b̄α(t)Yα = ξ(t) [(Y1 − 3Y3 − Y6) + f(t)Y9 + g(t)Y8 + h(t)(Y7 + Y4)] , (8.2) for a certain t-dependent non-vanishing function ξ(t). Taking into account that the vector fields Y1, . . . , Y11 are linearly independent over R, we see that b̄2 = 0, b̄5 = 0, b̄10 = 0, b̄11 = 0. Bearing in mind the form of the coefficients b̄α given by (4.3) and recalling that α > 0, we see that b̄2 = 0 entails n = 1. Meanwhile, conditions b̄5 = 0 and b̄11 = 0 entail σ = 0 and 1 α dα dt = γ δ , (8.3) respectively. In turn, σ = 0 also entails that b̄9 = b̄10 = 0. Moreover, from (8.2) we also see that b̄3 = −3b̄1, b̄6 = −b̄1, b̄7 = b̄4. From these conditions and n = 1, σ = 0, we obtain a0 α = −α δ , a20δ α3 = α δ , 1 δ dδ dt = 2γ δ − 1 a0 da0 dt . (8.4) Since δ, α > 0, it can readily be seen that the second equation is an immediate consequence of the first one, which in turn implies a0 = −α2/δ. Using that δ(0) = α(0) = 1, we obtain a0(0) = −1. Previous results along with (8.3) ensure that the last condition in (8.4) holds. Hence, b̄1, b̄4 and b̄8 become b̄1 = −a0 α , b̄4 = a1 − 1 a0 da0 dt + 3 α dα dt , A Quasi-Lie Schemes Approach to Second-Order Gambier Equations 19 b̄8 = −a2α a0 + a1 a0 dα dt + 2 a0α ( dα dt )2 − 1 a0 d2α dt2 , (8.5) and the remaining coefficients b̄α are either zero or can be obtained from them. Proposition 8.1. Every Gambier vector field X is a quasi-Lie system relative to the quasi-Lie scheme S(WG, V ′ G) and the Lie algebra V0 if and only if n = 1, a0(0) = −1 and σ = 0. In such a case, every t-dependent transformation g ∈ G(WG) given by x̄ = αx, v̄ = − α a0 ( dα dt x+ αv ) , (8.6) transforms X into a Lie system dx̄ dτ = v̄, dv̄ dτ = −3x̄v̄ − x̄3 + b̄4b̄ −1 1 x̄+ b̄8b̄ −1 1 ( x̄2 + v̄ ) , (8.7) where dτ = −a0dt/α and b̄1, b̄4 and b̄8 are given by (8.5). 9 Exact solutions for several second-order Gambier equations As a result of Proposition 8.1, we can apply to (8.7) the techniques of the theory of Lie systems so as to devise, by inverting the change of variables (8.6), new properties of the Gambier equations related to second-order Riccati equations. For instance, we obtain new exact solutions of some of these families of second-order Gambier equations. In the study of every Lie system, special relevance takes the algebraic structure of its Vessiot– Guldberg Lie algebras. When a Lie system possesses a solvable Vessiot–Guldberg Lie algebra, we can apply several methods to explicitly integrate it (cf. [9, 13, 19]). Otherwise, the general solution of the Lie system usually needs to be obtained through approximative methods [51] or expressed in terms of solutions of other Lie systems [12]. System (8.7) is a special case of a Lie system related to a Vessiot–Guldberg Lie algebra V0 iso- morphic to sl(3,R), which is simple and therefore difficult to integrate explicitly. Nevertheless, we can prove that (8.7) is in general related to a Vessiot–Guldberg Lie algebra that is a proper Lie subalgebra of V0. Moreover, we next prove that X is related to a solvable Vessiot–Guldberg Lie algebra in some particular cases that can be explicitly integrated. System (8.7) describes the integral curves of the t-dependent vector field Xt = Z1 + b̄4b̄ −1 1 Z2 + b̄8b̄ −1 1 Z3. For generic t-dependent functions b̄1, b̄4 and b̄8, the above system admits a Vessiot–Guldberg Lie algebra spanned by Z1 = X1, Z2 = (X3 + X7)/2, Z3 = (X8 − 2X4)/4 and their successive Lie brackets, which generates a proper Lie subalgebra V of V0. More specifically, we first have [Z1, Z2] = Z3 − Z4, [Z1, Z3] = −(Z1 + Z5)/2, [Z1, Z5] = Z6, where Z4 = X4, Z5 = X5, Z6 = X6. This shows that the smallest Lie algebra containing Z1, Z2 and Z3 must include these vector fields and Z4, Z5 and Z6. Since these vector fields additionally satisfy the following commutation relations [Z1, Z4] = Z5, [Z3, Z4] = 0, [Z1, Z6] = 0, [Z2, Z3] = Z2, [Z2, Z4] = −Z2, [Z2, Z5] = Z4 − Z3, [Z2, Z6] = Z5 − Z1, [Z3, Z5] = (Z5 + Z1)/2, [Z3, Z6] = Z6, [Z4, Z5] = −Z1, [Z4, Z6] = 0, [Z5, Z6] = 0, 20 J.F. Cariñena, P. Guha and J. de Lucas we see that they span a six-dimensional proper Lie subalgebra V of V0. Moreover, it is easy to show that V ' 〈Z1 +Z5, Z3−Z4, Z2〉 ⊕S 〈Z6, Z1−Z5, Z3 +Z4〉, where 〈Z1 +Z5, Z3−Z4, Z2〉 ' sl(2,R), namely [Z1 + Z5, Z3 − Z4] = −2(Z1 + Z5), [Z2, Z3 − Z4] = 2Z2, [Z1 + Z5, Z2] = 2(Z3 − Z4), the linear space 〈Z6, Z1 −Z5, Z3 +Z4〉 is a solvable ideal of V and ⊕S stands for the semidirect sum of 〈Z1 + Z5, Z3 − Z4, Z2〉 by 〈Z6, Z1 − Z5, Z3 + Z4〉. Hence, X is related to a non-solvable Vessiot–Guldberg Lie algebra and it is not known a general method to explicitly integrate X for arbitrary b̄1, b̄4 and b̄8. Nevertheless, we can focus on a particular instance of these functions so that X can be related to a solvable Vessiot– Guldberg Lie algebra V1. For example, consider the case b̄4 = 0. We then have V1 = 〈Z1, Z3, Z1 + Z5, Z6〉. Note that the derived series of V1 become zero, i.e. D1 ≡ [V1, V1] = 〈Z1 + Z5, Z6〉, D2 = 0. In other words, V1 is solvable and we can expect to integrate it through some method. Let us do so through the so-called mixed superposition rules, i.e. a generalisation of superposition rules describing the general solution of a Lie system in terms of several particular solutions of other systems and a set of constants [29]. In our case, it is known (cf. [29, p. 194]) that the general solution of (8.7) can be written in the form x̄(τ) = λ1vy1(τ) + λ2vy2(τ) + λ3vy3(τ) λ1y1(τ) + λ2y2(τ) + λ3y3(τ) , v̄(τ) = d dτ ( λ1vy1(τ) + λ2vy2(τ) + λ3vy3(τ) λ1y1(t) + λ2y2(τ) + λ3y3(τ) ) , where (λ1, λ2, λ3) ∈ R3 − {(0, 0, 0)} and (yi(τ), vyi(τ), ayi(τ)), with i = 1, 2, 3, are linearly independent solutions of the linear Lie system dy dτ = vy, dvy dτ = ay, day dτ = b̄8b̄ −1 1 ay. The above Lie system is related to a Vessiot–Guldberg Lie algebra spanned by W1 = vy ∂ ∂y + ay ∂ ∂vy , W2 = ay ∂ ∂ay , W3 = 2ay ∂ ∂vy , W4 = −2ay ∂ ∂y , which close the same commutation relations as Z1, Z3, Z1 + Z5 and Z6. Hence, this system is related to a solvable Vessiot–Guldberg Lie algebra and it can easily be integrated: ay = exp (∫ τ b̄8(τ ′)b̄−11 (τ ′)dτ ′ ) , vy = ∫ τ ay(τ ′)dτ ′, y = ∫ τ vy(τ ′)dτ ′. Since we can assume λ1y1(τ) + λ2y2(τ) + λ3y3(τ) = ∫ τ ∫ τ ′ exp (∫ τ ′′ b̄8(τ ′′′)b̄−11 (τ ′′′)dτ ′′′ ) dτ ′′dτ ′, we obtain the solution of (8.7) for b̄4 = 0, i.e. x̄(τ) = ∫ τ exp (∫ τ ′ b̄8(τ ′′)b̄−11 (τ ′′)dτ ′′ ) dτ ′∫ τ ∫ τ ′ exp (∫ τ ′′ b̄8(τ ′′′)b̄ −1 1 (τ ′′′)dτ ′′′ ) dτ ′′dτ ′ , v̄(τ) = dx̄ dτ (τ). From above results, we have the following proposition. A Quasi-Lie Schemes Approach to Second-Order Gambier Equations 21 Proposition 9.1. Given a second-order Gambier equation with n = 1, a0(0) = −1, σ = 0 and a particular solution α(t) with α(0) = 1 of a1 = 1 a0 da0 dt − 3 α dα dt , (9.1) its general solution reads x(τ(t)) = 1 α(τ(t)) ∫ τ(t) exp (∫ τ ′ b̄8(τ ′′)b̄−11 (τ ′′)dτ ′′ ) dτ ′∫ τ(t) ∫ τ ′ exp (∫ τ ′′ b̄8(τ ′′′)b̄ −1 1 (τ ′′′)dτ ′′′ ) dτ ′′dτ ′ , where dτ = −a0dt/α and b̄1 and b̄8 are given by (8.5). Note 9.1. Since equation (9.1) can be easily integrated, the above proposition shows that we can integrate exactly every second-order Gambier equation with n = 1, a0(0) = −1 and σ = 0. 10 Conclusions Two quasi-Lie schemes have been introduced to analyse the second-order Gambier equations. Our first quasi-Lie scheme has been used to recover previous results concerning the reduction of such equations to reduced canonical forms from a geometric clarifying approach, which allowed us to fill a gap in the previous literature. This quasi-Lie scheme also led to determine some quasi- Lie systems related to certain second-order Gambier equations, which enabled us to transform them into second-order Kummer–Schwarz equations. We have expressed the general solutions of such second-order Gambier equations in terms of particular solutions of t-dependent frequency harmonic oscillators and Riccati equations. Additionally, new constants of motion were derived for some of them. 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A Quasi-Lie Schemes Approach to Second-Order Gambier Equations

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