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

20 May 2020  ·  Blázquez-Sanz David, Casale Guy, Arboleda Juan Sebastián Díaz ·

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. read more

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Differential Geometry