Reversible linear differential equations

Let ∇ be a meromorphic connection on a vector bundle over a compact Riemann surface Γ. An automorphism σ:Γ→Γ is called a symmetry of ∇ if the pullback bundle and the pullback connection can be identified with ∇. We study the symmetries by means of what we call the Fano group; and, under the hypothesis that ∇ has a unimodular reductive Galois group, we relate the differential Galois group, the Fano group and the symmetries by means of an exact sequence.