Analytical embedding of three-dimensional Helmholtz-type geometries

For modern geometry, the study of maximum mobility geometries is important. The maximum mobility for $n$-dimensional geometry given by the function $f$ of a pair of points means the existence of an $n(n+1)/2$-dimensional transformation group, which leaves this function invariant. Many geometries of maximum mobility are known (Euclidean, symplectic, Lobachevsky, etc.), but there is no complete classification of such geometries. In this article, the method of embedding solves one of these classification problems. The essence of this method is as follows: from the function of a pair of points $ g $ of three-dimensional geometry, we find all non-degenerate functions $f$ of a pair of points of four-dimensional geometries that are invariants of the Lie group of transformations of dimension $10$. In this article, $g$ are non-degenerate functions of a pair of points of two Helmholtz three-dimensional geometries:
$$g = 2\ln(x_i-x_j) + \dfrac{y_i-y_j}{x_i-x_j} + 2z_i + 2z_j, $$

$$\ln [(x_i-x_j)^2 + (y_i-y_j)^2] + 2\gamma\,\mathrm{arctg}\,\dfrac{y_i-y_j}{x_i-x_j} + 2z_i + 2z_j. $$
These geometries are locally maximally mobile, that is, their groups of motions are six-dimensional. The problem solved in this work is reduced to solving special functional equations by analytical methods, the solutions of which are sought in the form of Taylor series. For searching various options, the math software package Maple 15 is used. As a result, only degenerate functions of a pair of points are obtained.