Two-dimensional Riemann–Hilbert problem for commutative monodromy on an elliptic curve

We obtain an explicit solution of the Riemann–Hilbert problem on an elliptic curve for the two-dimensional commutative monodromy representations. By an arbitrary set of points together with a representation of the fundamental group of the curve punctured at these points, we construct a semistable holomorphic vector bundle of degree zero with a logarithmic connection possessing the required singularities and monodromy.