On difference-differential Lax pairs and integrals of Painlevé equations in finite characteristic

Abstract

We collect rank two difference-differential Lax pairs for classical Painlevé equations in the literature and put each in $2\times 2$ matrix form with the coefficient matrix of the spectral equation a degree two matrix polynomial. We describe and apply a general method to obtain integrals of motion in characteristic $p$ from these Lax pairs. For every relevant Painlevé equation, this leads to a countable list of integrals of motion, with one entry for each prime $p$.