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This paper is dedicated to the differential Galois theory in the complexanalytic context for Lie-Vessiot systems. Those are the natural generaliza-tion of linear systems, and the more general class of differential equationsadimitting superposition laws, as recently stated in [5]. A Lie-Vessiot sys-tem is automatically translated into a equation in a Lie group that we callautomorphic system. Reciprocally an automorphic system induces a hierarchy ofLie-Vessiot systems. In this work we study the global analytic aspects of aclassical method of reduction of differential equations, due to S. Lie. Wepropose an differential Galois theory for automorphic systems, and explore therelationship between integrability in terms of Galois the- ory and the Lie'sreduction method. Finally we explore the algebra of Lie symmetries of a generalautomorphic system.

Lie's Reduction Method and Differential Galois Theory in the Complex Analytic Context