The Malgrange–Galois groupoid of the Painlevé VI equation with parameters

Abstract

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 the Painlevé VI family including all parameters as new dependent variables. We conclude that it is the pseudogroup of transformations preserving the parameter values, the differential of the independent variable, a volume form in the dependent variables and the equation. This implies that a solution of a Painlevé VI equation depending analytically on the parameters does not satisfy any new partial differential equation (including derivatives with respect to parameters) which is not derived from Painlevé VI.