# On connections of the Li\'enard equation with some equations of Painlev\'e--Gambier type

## TL;DR

This paper explores the links between the Lie9nard equation and Painleve9--Gambier equations, enabling the construction of new analytical solutions and three integrable families of the Lie9nard equation.

## Contribution

It introduces connections between Lie9nard and Painleve9--Gambier equations, leading to new integrable families and a method for finding analytical solutions.

## Key findings

- Identified three new integrable Lie9nard families.
- Developed an approach for one-parameter analytical solutions.
- Established connections enabling solution construction.

## Abstract

The Li\'enard equation is used in various applications. Therefore, constructing general analytical solutions of this equation is an important problem. Here we study connections between the Li\'enard equation and some equations from the Painlev\'e--Gambier classification. We show that with the help of such connections one can construct general analytical solutions of the Li\'enard equation's subfamilies. In particular, we find three new integrable families of the Li\'enard equation. We also propose and discuss an approach for finding one--parameter families of closed--form analytical solutions of the Li\'enard equation.