On the Riemann-Hilbert problem with a piecewise constant matrix

A constructive solution to the Riemann-Hilbert boundary value problem as well as to the C-linear conjugation problem (or Riemann problem) with a special piecewise constant matrix for a multiply connected domain is obtained. The result is based on a vector generalization of the scalar method of functional equations in complex domain and on application of the Christoffel-Schwarz transformation of the half-plane onto a circular polygon.