Motion group of the simplicial plane as a solution of a functional equation

Proceeding from metrical viewing of geometry, which appeared in the 19th century in works of H. Helmholtz and A. Poincare and which is deeply connected with F. Klein’s group conception, Yu. I. Kulakov created a general conception of distance — the physical structures' theory based on the so-called phenomenological symmetry. The essence of this symmetry is the fact that there is a functional connection between all mutual distances for $n+2$ arbitrary points in the $n$-dimensional space. G. G. Mikhailichenko’s works established the equivalence of the group and phenomenological symmetries, which helped him to construct a complete classification of two- dimensional phenomenologically symmetric geometries. Along with well-known two- dimensional geometries (Euclidean plane, Lobachevsky plane, Minkowski plane, symplectic plane, two-dimensional sphere, and two-dimensional one-sheet hyperboloid) this classification shows the Helmholtz plane, pseudo-Helmholtz plane, and simplicial plane which was also the object of study of such geometricians as A. A. Aleksandrov and R. I. Pimenov.
The aim of this study is to find a local group of the set of all simplicial plane motions as a solution for a functional equation.
Defining the set of plane motions preserving the metric function as a function of a pair of points leads to developing analytical methods for solving the corresponding functional equations; that allows one to complement the theory of functional equations since there are few general methods of their solution.
It has been found that any motion of a simplicial plane is defined by a linear transformation, which was not assumed beforehand and was not obvious. Nevertheless the whole set of motions turned out to be a group essentially dependent on three independent parameters. Thus, a simplicial plane is endowed with the group symmetry of the 3rd degree, i.e. it is a phenomenologically symmetric geometry with maximum mobility. It should be noted that this result is not valid for an arbitrary geometry but is typical just for phenomenologically symmetric geometries.