  In this paper we develop a differential Galois theory for algebraic
Lie-Vessiot systems in algebraic homogeneous spaces. Lie-Vessiot systems are
non autonomous vector fields that are linear combinations with time-dependent
coefficients of fundamental vector fields of an algebraic Lie group action.
Those systems are the building blocks for differential equations that admit
superposition of solutions. Lie-Vessiot systems in algebraic homogeneous spaces
include the case of linear differential equations. Therefore, the differential
Galois theory for Lie-Vessiot systems is an extension of the classical
Picard-Vessiot theory. In particular, algebraic Lie-Vessiot systems are
solvable in terms of Kolchin's strongly normal extensions. Therefore, strongly
normal extensions are geometrically interpreted as the fields of functions on
principal homogeneous spaces over the Galois group. Finally we consider the
problem of integrability and solvability of automorphic differential equations.
Our main tool is a classical method of reduction, somewhere cited as Lie
reduction. We develop and algebraic version of this method, that we call
Lie-Kolchin reduction. Obstructions to the application are related to Galois
cohomology.
