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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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/

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