# The Lie symmetry group of the general Lienard-type equation

## TL;DR

This paper characterizes the conditions under which a general Lienard-type differential equation admits additional Lie symmetries beyond the obvious time translation symmetry, extending previous results to higher-order cases.

## Contribution

It provides a complete characterization and verifiable conditions for extra Lie symmetries in Lienard-type equations for all orders except three.

## Key findings

- Identifies when additional Lie symmetries occur
- Provides an easily checkable condition on the functions f_k
- Extends symmetry analysis to higher-order Lienard equations

## Abstract

We consider the general Lienard-type equation $\ddot{u} = \sum_{k=0}^n f_k \dot{u}^k$ for $n\geq 4$. This equation naturally admits the Lie symmetry $\frac{\partial}{\partial t}$. We completely characterize when this equation admits another Lie symmetry, and give an easily verifiable condition for this on the functions $f_0, \dots , f_n$. Moreover, we give an equivalent characterization of this condition. Similar results have already been obtained previously in the cases $n=1$ or $n=2$. That is, this paper handles all remaining cases except for $n=3$.