Special Monodromy Groups and the Riemann–Hilbert Problem for the Riemann Equation

In this paper, we solve the Riemann–Hilbert problem for the Riemann equation and for the hypergeometric equation. We describe all possible representations of the monodromy of the Riemann equation. We show that if the monodromy of the Riemann equation belongs to $SL(2,\mathbb C)$, then it can be realized not only by the Riemann equation, but also by the more special Riemann–Sturm–Liouville equation. For the hypergeometric equation, we construct a criterion for its monodromy group to belong to $SL(2,\mathbb Z)$.