Reversible linear differential equations (Q630854)

In 1900, \textit{G. Fano} [Math. Ann. 53, 493--590 (1900; JFM 31.0342.01)] has asked the question whether or not a linear differential equation with solutions satisfying a homogeneous polynomial can be solved in terms of linear equations of lower order. The author treats the problem in terms of connections. Let \(\nabla\) be a meromorphic connection on a vector bundle over a compact Riemann surface \(\Gamma \). An automorphism \(\sigma :\varGamma \rightarrow \varGamma \) is called a symmetry of \(\nabla\) if the pullback bundle and the pullback connection can be identified with \(\nabla\). The author studies the symmetries by means of what he calls the Fano group; and, under the hypothesis that \(\nabla\) has a unimodular reductive Galois group, he relates the differential Galois group, the Fano group and the symmetries by means of an exact sequence.