The master Painlev\'e VI heat equation

Given the second order scalar Lax pair of the sixth Painlev'e equation, we build a generalized heat equation with rational coefficients which does not depend any more on the Painlev'e variable. 'Etant donn'e la paire de Lax scalaire de la sixi`eme 'equation de Painlev'e, nous donnons une construction directe de l’'equation de la chaleur g'en'eralis'ee `a coefficients rationnels qui ne d'epend plus de la variable de Painlev'e.