On difference-differential Lax pairs and integrals of Painlev\'e equations in finite characteristic

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 $p$.