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The boundary value problems for Poisson's equation in the plane domains represented by wedges and layers are considered. Conditions of a general form along all the interior and exterior boundaries are prescribed. The analysis is significantly simplified by incorporating the geometrical features of the layers and wedges: they present chain-like systems. The essence of the method applied consists in using the Fourier and Mellin transforms for the corresponding regions, and in combining the transformations of respective functions along the common boundaries. The problems are reduced to systems of functional or functional-difference equations, and later to systems of singular integral equations with fixed point singularities. The results, concerning the solvability of the obtained systems of the integral equations are presented. In the Appendix the formulae are also given making it possible to use directly the results obtained from this and the previous paper to solve the boundary value problems for linear partial-differential equations of divergence form in a similar domain, corresponding to physical problems for anisotropic nonhomogeneous bodies.