On the embedding of two-dimetric phenomenologically symmetric geometries

The two-dimetric phenomenologically symmetric geometry of two sets (TPS GTM) of rank $(n + 1, 2)$, where $n = 1, 2,\dots$, is defined on a two-dimensional and $2n$-dimensional differentiable manifolds $M$ and $N$ by a differentiable nondegenerate function $f: M \times N \to R^2$ with an open and dense domain and the axiom of phenomenological symmetry. There is a complete classification of the TPS GTM of rank $(n + 1, 2)$, and the functions that define these geometries are locally isotopic to $n$-transitive actions of certain Lie groups on a two-dimensional manifold. From this classification, it can be seen that functions of some TPS GTM of rank $(n + 1, 2)$ contain functions of the TPS GTM of rank $(n, 2)$ as an argument.
In this paper, we introduce the definition of an embedding according to which the TPS GTM of rank $(n, 2)$, given by the function $g = (g^1, g^2)$, is embedded in the TPS GTM of rank $(n + 1, 2)$ with the function $f = (f^1, f^2)$ if the function f contains the function $g$ as an argument. The problem is to find the embeddings for the TPS GTM of rank $(n + 1, 2)$. As a result, an important theorem is proved, according to which at least one of the TPS GTM of rank $(n, 2)$, where $n = 2, 3, 4$, is embedded in each of the TPS GTMs of rank $(n + 1, 2)$. The problem is solved by the group method and is reduced to distinguishing the stationary subgroups of the transformation groups to which the previously known TPS GTMs are locally isotopic. In the process of proving the theorem, it is established that the transformation group defining the TPS GTM of rank $(n + 1, 2)$ is a composition of the stationary subgroup defining the TPS GTM of rank $(n, 2)$ and some subgroup. It is also proved that transformation groups that are locally isotopic to a TPS GTM of rank $(n + 1, 2)$ are nearly $n$-transitive. The last property means that parameters of such a group of transformations can be expressed in terms of coordinates of a certain number of points.