Malgrange-Galois groupoid of Painlev\'e VI equation with parameters

TL;DR

This paper computes the Malgrange-Galois groupoid of the Painlevé VI equation with parameters, showing it preserves specific geometric structures and implying solutions do not satisfy additional PDEs involving parameters.

Contribution

It extends the computation of the Malgrange-Galois groupoid to Painlevé VI with parameters, revealing its structure and implications for solution properties.

Findings

The Galois groupoid preserves parameter values and geometric structures.

Solutions depend analytically on parameters without satisfying new PDEs.

The result generalizes previous work on Painlevé IV.

Abstract

The Malgrange-Galois groupoid of Painlev\'e IV equations is known to be, for very general values of parameters, the pseudogroup of transformations of the phase space preserving a volume form, a time form and the equation. Here we compute the Malgrange-Galois groupoid of Painlev\'e VI family including all parameters as new dependent variables. We conclude it is the pseoudogroup of transformations preserving parameter values, the differential of the independent variable, a volume form in the dependent variables and the equation. This implies that a solution of Painlev\'e VI depending analytically on parameters does not satisfy any new partial differential equation (including derivatives w. r. t. parameters) which is not derived from Painlev\'e VI.