Classification of two-metric phenomenologically symmetric two-dimentional geometries of rang 3

Two-metric phenomenologically symmetric two-dimensional geometry are set on a two-dimensional variety $\mathrm M_2$ by a function of a couple of points $f(i,j)$ which we call the metric function. One of defining properties of this geometry is so-called phenomenological symmetry to which the special attention was paid for the first time by Yu. I. Kulakov who made it the basic principle of his theory of physical structures. The essence of phenomenological symmetry is that, in an $sn$-dimensional space between all mutual $sm(m-1)/2$ distances (that is used as the metric function) for $m=n+2$ arbitrary points, there is a functional relation. In the first part of this work, main definitions and axioms used further in the proof of theorems are given. In the second part of work presents a classification of two-metric phenomenologically symmetric two-dimensional geometries of rank 3 ($s=2$ and $n=1$) carried out by the analytical method in which the geometrical meaning of the studied functional and differential relations is used: first of all, a smooth replacement of local coordinates which allows one to reduce solutions of the differential equations to the simplest form. The specified reduction of solutions of the differential equations allows one to compare our solutions with results of other authors.