Riemann Problem and Singular Integral Equationswith Coefficients Generated by PiecewiseConstant Functions

We study a Riemann boundary value problem with a shift into the interior of the domain. The problem has piecewise constant coefficients that take two values. We find conditionsfor the existence and uniqueness of a solution of the inhomogeneous problem and formulas for the number of linearly independent solutions of the homogeneous problem.We consider scalar singular integral operators with a shift and matrix characteristic operators whose coefficients are generated by piecewise constant functions and which have automorphic properties. For these operators, we find invertibility conditions.