On the integrability conditions for a family of the Li\'enard-type equations

TL;DR

This paper establishes new integrability conditions for a family of Lie9nard-type equations using nonlocal transformations and connections to Painleve9-Gambier equations, providing new criteria and examples of integrable cases.

Contribution

It introduces four new integrability criteria for Lie9nard-type equations by linking them to Painleve9-Gambier equations and exploring nonlocal transformations.

Findings

Four new integrability criteria derived.

Examples of integrable Lie9nard equations provided.

Relationships between linearizability and integrability discussed.

Abstract

We study a family of Li\'enard--type equations. Such equations are used for the description of various processes in physics, mechanics and biology and also appear as traveling--wave reductions of some nonlinear partial differential equations. In this work we find new conditions for the integrability of this equations family. To this end we use an approach, which is bases on application of nonlocal transformations. By studying connections between this family of Li\'enard--type equations and type III Painlev\'e--Gambier equations, we obtain four new integrability criteria. We illustrate our results by providing examples of some integrable Li\'enard--type equations. We also discuss relationships between linearizability via nonlocal transformations of this family of Li\'enard--type equations and other integrability conditions for this family of equations.