Singularity analysis of a variant of the Painlevé–Ince equation

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