Embedding of an additive two-dimensional phenomenologically symmetric geometry of two sets of rank $(2,2)$ into two-dimensional phenomenologically symmetric geometries of two sets of rank $(3,2)$

In this paper, the method of embedding is used to construct the classification of two-dimensional phenomenologically symmetric geometries of two sets (PS GTS) of rank $(3,2)$ from the previously known additive two-dimensional PS GTS of rank $(2,2)$ defined by a pair of functions $g^1=x+\xi$ and $g^2 = y+\eta$. The essence of this method consists in finding the functions defining the PS GTS of rank $(3,2)$ with respect to the functions $g^1=x+\xi$ and $g^2 = y+\eta$. In solving this problem, we use the fact that the two-dimensional PS GTS of rank $(3,2)$ admit groups of transformations of dimension 4, and the two-dimensional PS GTS of rank $(2,2)$ is of dimension $2$. It follows that the components of the operators of the Lie algebra of the transformation group of the two-dimensional PS GTS of rank $(3,2)$ are solutions of a system of eight linear differential equations of the first order in two variables. Investigating this system of equations, we arrive at possible expressions for systems of operators. Then, from the systems of operators, we select the operators that form Lie algebras. Then, applying the exponential mapping, we recover the actions of the Lie groups from the Lie algebras found. It is precisely these actions that specify the two-dimensional PS GTS of rank $(3,2)$.