Lax pairs and first integrals for autonomous and non-autonomous differential equations belonging to the painlevé – Gambier list

Recently Sinelshchikov et al. [1] formulated a Lax representation for a family of non-autonomous 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.