Examples of effective solution of the Riemann-Hilbert problem on renewal of a differential equation with monodromy group in the framework of the theory of automorphic functions

In this paper we study the classical problem of reconstructing a differential equation from a given monodromy group $\Gamma$ in the situation when $\Gamma$ is a Fuchsian group of the first kind of topological genus zero. We also give a survey of the algebrotopological theory which describes the structure of the subgroups of $\Gamma$, and we give examples of the calculation of the corresponding modular equations.