Lie point symmetries and ODEs passing the Painlevé test

Abstract

The Lie point symmetries of ordinary differential equations (ODEs) that are candidates for having the Painlevé property are explored for ODEs of order $n =2, \dots ,5$. Among the 6 ODEs identifying the Painlevé transcendents only $P_{III}$, $P_V$ and $P_{VI}$ have nontrivial symmetry algebras and that only for very special values of the parameters. In those cases the transcendents can be expressed in terms of simpler functions, i.e. elementary functions, solutions of linear equations, elliptic functions or Painlevé transcendents occurring at lower order. For higher order or higher degree ODEs that pass the Painlevé test only very partial classifications have been published. We consider many examples that exist in the literature and show how their symmetry groups help to identify those that may define genuinely new transcendents.