A truncation for obtaining all the first degree birational transformations of the Painlev\'e transcendents

Abstract

A birational transformation is one which leaves invariant an ordinary differential equation, only changing its parameters. We first recall the consistent truncation which has allowed us to obtain the first degree birational transformation of Okamoto for the master Painlev\'e equation P6. Then we improve it by adding a preliminary step, which is to find all the Riccati subequations of the considered Pn before performing the truncation. We discuss in some detail the main novelties of our method, taking as an example the simplest Painlev\'e equation for that purpose, P2. Finally, we apply the method to P5 and obtain its two inequivalent first degree birational transformations.