Construction of 2× 2 Fuchsian System with Five Singular Points

A constructive method is proposed for the solution of the Riemann–Hilbert problem which is realized in the case of five singular points. The basic components of this method are the logarithmization of the product of matrix functions and the determination of the parameters of a related differential equation of Fuchsian type. The proposed method allows us to solve several related problems, namely, to factorize the piecewise constant matrix functions, to find partial indices and to explicitly solve the vector-matrix boundary value problem with the above piecewise constant matrix function as a coefficient.