Lie point symmetries and ODEs passing the Painlev\'e test

TL;DR

This paper investigates the Lie point symmetries of second to fifth order ODEs that pass the Painlevé test, revealing how symmetry analysis can identify equations defining new transcendents or expressible solutions.

Contribution

It provides a detailed symmetry analysis of Painlevé test passing ODEs, highlighting the special cases with nontrivial symmetries and their implications for solutions.

Findings

Only specific Painlevé equations have nontrivial symmetry algebras.

Symmetry groups help identify equations with solutions in elementary functions or lower-order transcendents.

Many higher-order Painlevé test passing ODEs lack complete symmetry classifications.

Abstract

The Lie point symmetries of ordinary differential equations (ODEs) that are candidates for having the Painlev\'e property are explored for ODEs of order $n =2, \dots ,5$. Among the 6 ODEs identifying the Painlev\'e 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 Painlev\'e transcendents occurring at lower order. For higher order or higher degree ODEs that pass the Painlev\'e 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.