On the criteria for integrability of the Li\'{e}nard equation

TL;DR

This paper introduces a new criterion for determining the integrability of the Lie9nard equation using nonlocal transformations, unifying and extending previous criteria with new examples.

Contribution

The paper presents a novel approach based on nonlocal transformations to establish integrability criteria for the Lie9nard equation, including new integrable cases.

Findings

Derived a new integrability criterion for the Lie9nard equation.

Reproduced known criteria as special cases of the new approach.

Provided several new examples of integrable Lie9nard equations.

Abstract

The Li\'{e}nard equation is of a high importance from both mathematical and physical points of view. However a question about integrability of this equation has not been completely answered yet. Here we provide a new criterion for integrability of the Li\'{e}nard equation using an approach based on nonlocal transformations. We also obtain some of previously known criteria for integrability of the Li\'{e}nard equation as a straightforward consequences of our approach's application. We illustrate our results by several new examples of integrable Li\'{e}nard equations.