Rational ODEs with Movable Algebraic Singularities

Abstract

A class of second‐order rational ordinary differential equations, admitting certain families of formal algebraic series solutions, is considered. For all solutions of these equations, it is shown that any movable singularity that can be reached by analytic continuation along a finite‐length curve is an algebraic branch point. The existence of these formal series expansions is straightforward to determine for any given equation in the class considered. We apply the theorem to a family of equations, admitting different kinds of algebraic singularities. As a further application we recover the known fact for generic values of parameters that the only movable singularities of solutions of the Painlevé equations PII–P VI are poles.