On one system of functional equations for embedding an additive rank (2, 2) into a dual rank (3, 2) of two-metric phenomenologically symmetric geometries of two sets

Systems of two functional equations with several unknown functions of several variables naturally appear when establishing the mutual embedding of two-metric phenomenologically symmetric geometries of two sets (TPS GTS). An embedding is possible if the corresponding system of functional equations has at least one nondegenerate solution. Two-metric phenomenologically symmetric geometries were studied previously in the works of G.G. Mihailichenko and R.A. Bogdanova within the framework of the problems of constructing and classifying such geometries arising in a more general theory of phenomenologically symmetric geometries - geometries of maximum mobility. The discovery made in the beginning of the 19th century by Gauss, Lobachevsky, and Bolyai that Euclidean geometry is not the only possible one made it possible in the 20th century, along with other geometries, to discover geometries of maximum mobility, representing a separate class of geometries that admit a maximum group of movements. The origin of this theory in the 1960s was associated with the tasks of mathematical substantiation of the theory of relativity and other classical laws of physics. Phenomenologically symmetric geometries, which are geometries of local maximum mobility, represent a synthesis of two classical approaches to the construction of geometry: the group and metric approaches which for many decades (starting with the works of G. Helmholtz, F. Klein, A. Poincare, S. Lee, A. Cayley, etc.) served as a tool for research in the theory of representations of Lie groups, Riemannian geometry, and other branches of mathematics. Researchers who adhere to this direction associate a pair of points with not one value of a two-point function (a function of a pair of points)-an invariant of the corresponding transformation group - but several. In this paper, it is proposed to develop a method for finding a general nondegenerate solution of a system of two functional equations corresponding to embedding of an additive rank (2, 2) TPS of GTS into a dual rank (3, 2) TPS of GTS, which is an interesting and meaningful problem in the mathematical sense. This method can be developed and applied to other similar systems of functional equations arising within the framework of the problem of embedding TPS GTS.

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