# Singularity Analysis of a Variant of the Painlev{\'e}--Ince Equation

## TL;DR

This paper analyzes a modified Painlevé--Ince equation derived from the Euler--Bernoulli Beam equation, revealing its singularity structure and proposing a conjecture about the existence of Left Painlevé Series.

## Contribution

It introduces a new variant of the Painlevé--Ince equation, examines its singularity properties, and conjectures about the conditions for Left Painlevé Series in ODEs.

## Key findings

- The equation shares the same leading-order behaviour and resonances as the Painlevé--Ince Equation.
- It exhibits a Right Painlevé Series but no Left Painlevé Series.
- A conjecture regarding the existence of Left Painlevé Series is proposed.

## Abstract

We examine by singularity analysis an equation derived by reduction using Lie point symmetries from the Euler--Bernoulli Beam equation which is the Painlev\'{e}--Ince Equation with additional terms. The equation possesses the same leading-order behaviour and resonances as the Painlev\'{e}--Ince Equation and has a Right Painlev\'{e} Series. However, it has no Left Painlev% \'{e} Series. A conjecture for the existence of Left Painlev\'{e} Series for ordinary differential equations is given.