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Samarskii-Ionkin type problems are known classes of problems that represent a generalization of classical ones. At the same time they are obtained in a natural way by constructing mathematical models of real processes and phenomena in physics, engineering, sociology, ecology, etc. Here we investigate the ability to solve non-local problems of its type in 2D using the Fourier method of the separation of variables. We study the completeness of the root functions of the corresponding spectral problems in L2(0 < x, y < 1), when they are defined as products of two systems of functions, where one of them is an orthonormal basis, and another is a Riesz basis. Using the properties of biorthogonal systems, we also study the problem of identifying the source function in the spatial domain.

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