The master Painlev\'e VI heat equation

Abstract

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.