The Riemann–Hilbert problem in Hardy classes for general first-order elliptic systems

We consider the Riemann–Hilbert (Hilbert) problem in casses similar to the Hardy class for general first-order elliptic systems on a plane. We establish basic properties of Hardy classes for solutions to such systems and conditions of solvability of boundary-value problem. We construct the example that demonstrates that in the case of discontinuous coefficient of the boundary condition the picture of solvability can be different from the picture of solvability of the problem for holomorphic and generalized analytic functions. In particular, the problem can be unsolvable when its index is positive.