# A Systematic Analysis of the Properties of the Generalised Painlev\'e--Ince Equation

## TL;DR

This paper analyzes the generalized Painleve-Ince equation, exploring its symmetry properties and singularity structure, revealing conditions for integrability and symmetry, and demonstrating its behavior under the Painleve test.

## Contribution

It provides a detailed symmetry and singularity analysis of the generalized Painleve-Ince equation, identifying conditions for maximal symmetry and integrability.

## Key findings

- Maximal symmetry when eta = lpha^2/9
- Fails the Painleve test for arbitrary parameters
- Becomes integrable under symmetry and singularity analysis

## Abstract

We consider the generalized Painlev\'e--Ince equation, \begin{equation*} \ddot{x}+\alpha x\dot{x}+\beta x^{3}=0 \end{equation*} and we perform a detailed study in terms of symmetry analysis and of the singularity analysis. When the free parameters are related as $\beta =\alpha ^{2}/9~$the given differential equation is maximally symmetric and well-known that it pass the Painlev\'{e} test. For arbitrary parameters we find that there exists only two Lie point symmetries which can be used to reduce the differential equation into an algebraic equation. However, the generalized Painlev\'{e}--Ince equation fails at the Painlev\'{e} test, except if we apply the singularity analysis for the new second-order differential equation which follows from the change of variable $x=1/y.$ We conclude that the Painlev\'{e}--Ince equation is integrable is terms of Lie symmetries and of the Painlev\'{e} test.