Discrete point symmetries of ordinary differential equations

Abstract

This paper describes a method that enables the user to construct systematically the set of all discrete point symmetries of a given ordinary differential equation (ODE) of order two or greater, provided that the ODE has at least a one–parameter Lie group of point symmetries. The method is easy to use, and is based upon Li's method of constructing continuous symmetries. The calculations are simple, and computer algebra is not usually required. Various examples are used to illustrate the method. The paper concludes with a proof that every ODE whose Lie group of point symmetries is isomorphic to the unimodular group has at least four inequivalent real discrete point symmetries.