  We study the interplay between the differential Galois group and the Lie
algebra of infinitesimal symmetries of systems of linear differential
equations. We show that some symmetries can be seen as solutions of a hierarchy
of linear differential systems. We show that the existence of rational
symmetries constrains the differential Galois group in the system in a way that
depends of the Maclaurin series of the symmetry along the zero solution.
