Malgrange-Galois groupoid of Painlevé VI equation with parameters
arXiv:2005.10291
Abstract
The Malgrange-Galois groupoid of Painlevé 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 Painlevé 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 Painlevé VI depending analytically on parameters does not satisfy any new partial differential equation (including derivatives w. r. t. parameters) which is not derived from Painlevé VI.
arXiv comment35 pages. Revised version