Geodesic
Lax Pairs and First Integrals for Autonomous and Non-Autonomous Differential Equations Belonging to the Painlevé – Gambier List
Russian journal of nonlinear dynamics, Tome 16 (2020) no. 4, pp. 637-650

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Recently Sinelshchikov et al. [1] formulated a Lax representation for a family of nonautonomous second-order differential equations. In this paper we extend their result and obtain the Lax pair and the associated first integral of a non-autonomous version of the Levinson – Smith equation. In addition, we have obtained Lax pairs and first integrals for several equations of the Painlevé – Gambier list, namely, the autonomous equations numbered XII, XVII, XVIII, XIX, XXI, XXII, XXIII, XXIX, XXXII, XXXVII, XLI, XLIII, as well as the non-autonomous equations Nos. XV and XVI in Ince’s book.
Keywords: Lax representation, first integrals.
Mots-clés : Liénard type equations, Painlevé – Gambier equations 
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     author = {P. Guha and S. Garai and A. G. Choudhury},
     title = {Lax {Pairs} and {First} {Integrals} for {Autonomous} and {Non-Autonomous} {Differential} {Equations} {Belonging} to the {Painlev\'e} {\textendash} {Gambier} {List}},
     journal = {Russian journal of nonlinear dynamics},
     pages = {637--650},
     publisher = {mathdoc},
     volume = {16},
     number = {4},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ND_2020_16_4_a7/}
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P. Guha; S. Garai; A. G. Choudhury. Lax Pairs and First Integrals for Autonomous and Non-Autonomous Differential Equations Belonging to the Painlevé – Gambier List. Russian journal of nonlinear dynamics, Tome 16 (2020) no. 4, pp. 637-650. http://geodesic.mathdoc.fr/item/ND_2020_16_4_a7/