# Lax representation and quadratic first integrals for a family of non-autonomous second-order differential equations

## TL;DR

This paper establishes a link between quadratic first integrals and Lax representations for a family of non-autonomous second-order differential equations, extending integrability concepts to dissipative systems.

## Contribution

It provides necessary and sufficient conditions for these equations to admit quadratic first integrals and Lax representations, connecting integrability properties in dissipative systems.

## Key findings

- Conditions for quadratic first integrals are equivalent to Lax representations.
- Examples include generalizations of Van der Pol and Duffing equations.
- The results extend integrability concepts to non-Hamiltonian dissipative systems.

## Abstract

We consider a family of non-autonomous second-order differential equations, which generalizes the Li\'enard equation. We explicitly find the necessary and sufficient conditions for members of this family of equations to admit quadratic, with the respect to the first derivative, first integrals. We show that these conditions are equivalent to the conditions for equations in the family under consideration to possess Lax representations. This provides a connection between the existence of a quadratic first integral and a Lax representation for the studied dissipative differential equations, which may be considered as an analogue to the theorem that connects Lax integrability and Arnold--Liouville integrability of Hamiltonian systems. We illustrate our results by several examples of dissipative equations, including generalizations of the Van der Pol and Duffing equations, each of which have both a quadratic first integral and a Lax representation.