Embedding of phenomenologically symmetric geometries of two sets of the rank $(N,2)$ into phenomenologically symmetric geometries of two sets of the rank $(N+1,2)$

In this paper, we propose a new method of classification of metric functions of phenomenologically symmetric geometries of two sets. It is called the method of embedding, the essence of which is to find the metric functions of phenomenologically symmetric geometries of two high-rank sets for the given phenomenologically symmetric geometry of two sets having rank less by 1. By the previously known metric function of phenomenologically symmetric geometry of two sets of the rank $(2,2)$ the metric function of phenomenologically symmetric geometry of two sets of the rank $(3,2)$ is found, by the phenomenologically symmetric geometry of two sets of the rank $(3,2)$ we find phenomenologically symmetric geometry of two sets of the rank $(4,2)$. Then it is proved that embedding of phenomenologically symmetric geometry of two sets of the rank $(4,2)$ into the phenomenologically symmetric geometry of two sets of the rank $(5,2)$ is absent. To solve the problem we generate special functional equations which are reduced to well-known differential equations.