A two-dimensional Riemann-Hilbert problem on an elliptic curve

A generalization of the Riemann–Hilbert problem to Riemann surfaces of positive genus means a question on the existence of a semistable vector bundle of degree zero over a surface, with a logarithmic connection having a given set of singular points and a given monodromy representation of the fundamental group of a punctured surface. We consider the two-dimensional problem with tree singular points on an elliptic curve. In this case a bundle and a connection can be constructed explicitely. It turns out that irreducible monodromy representations are realized for an arbitrary location of singular points, whereas for the realization of reducible representations some additional relations on the monodromy data, location of singular points and the spectral parameter of a bundle are required.