For the purposes of evaluating the behavior of multicomponent materials under operating conditions, the paper for the first time constructs in general an exact solution of systems of Wiener-Hopf integral equations of arbitrary order. Systems of these equations arise in mixed problems of continuum mechanics for multicomponent materials of complex rheology. The cases of mixed boundary value problems are considered under the assumption that there is a change of boundary conditions on the inner or outer boundary of a multilayer medium. Such mixed problems are reduced to systems of Wiener-Hopf integral equations, the Fourier transform of the kernels of which is a fairly general meromorphic matrix-a function that does not coincide with any of the special cases for which the system of equations is precisely solved. Earlier, the authors considered the case of a system consisting of two equations. The transfer of these results to the case of an arbitrary number of equations is based on this previously performed work. Systems of Wiener-Hopf integral equations arise in the mechanics of deformable media, geophysics, flaw detection, economics, and in a number of related fields.