Two-point invariants of groups of motions in some phenomenologically symmetric two-dimensional geometries

In G.G. Mikhaylichenko's classification, along with the well-known geometries, such as the Euclidean plane, Minkowsky plane, two-dimensional sphere, and others, there are two-dimensional Helmholtz type geometries in which the circle does not have the usual pattern, as evidenced by Helmholtz in his work “On the Facts Underlying Geometry”, as well as the simplicial plane. All these geometries are endowed by group and phenomenological symmetries. The essence of the phenomenological symmetry is in the link between all the mutual distances for a finite number of points.
The paper describes a complete system of non-degenerate two-point invariants of groups of motions for some phenomenologically symmetric two-dimensional geometries (Helmholtz plane, pseudo-Helmholtz plane, dual-Helmholtz plane, and simplicial plane) as a solution of corresponding functional equations for a set of two-point invariants of transformation groups.
The paper found that every two-point invariant of motion groups of the aforementioned geometries coincides with the metric function of the corresponding plane up to a smooth transformation $\psi(f)\to f$.