Lax representation and a quadratic rational first integral for second-order differential equations with cubic nonlinearity

In this paper we give a Lax formulation for a family of non-autonomous second-order differential equations of the type yzz + a3(z, y)y3z + a2(z, y)y2 z + a1(z, y)yz + a0(z, y) = 0. We obtain a sufficient condition for the existence of a Lax representation with a certain L-matrix. We demonstrate that equations with this Lax representation possess a quadratic rational first integral. We illustrate our construction with an example of the Rayleigh-Duffing-Van der Pol oscillator with quadratic damping. (c) 2022 Elsevier B.V. All rights reserved.