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