Analytic embedding of geometries of constant curvature on a pseudosphere

In mathematical studies, the geometries of maximum mobility are important. Examples of such geometries are Euclidean, pseudo-Euclidean, Lobachevsky, symplectic and so on. There is no complete classification of such geometries. They are distinguished as the geometries of the maximum mobility in general, for example, the geometries from the Thurston list, and the geometries of the local maximum mobility. V. A. Kyrov developed a method for classifying the geometries of local maximum mobility, called the method of embedding. The primary purpose of this paper is to find the metric functions of geometries of dimension $n + 2$ that admit $(n + 2)(n + 3)/2$ -parametric group of motions, and as an argument contain the metric function
$$ g(i,j) = \dfrac{\varepsilon_1(x^1_i - x^1_j)^2 + \cdots + \varepsilon_n(x^n_i - x^n_j)^2 + \varepsilon((x^{n+1}_i)^2 + (x^{n+1}_j)^2)}{x^{n+1}_ix^{n+1}_j} $$
of $(n + 1)$-dimensional geometry of constant curvature on a pseudosphere. In solving this problem, a functional equation of a special form is written due to the requirement for the existence of a group of motions of dimension $ (n + 2) (n + 3) / 2$, that is, of a group of transformations that preserve the metric function. When solving this problem with the requirement that a group of motions of dimension $ (n + 2) (n + 3) / 2$ exists, a functional equation of a special form can be written for this function. This functional equation is solved analytically, that is, all the functions are represented as Taylor series, then the coefficients in the expansions are compared. The result of solving this problem is the geometry of maximum mobility with the metric function
$$ f(i,j) = [\varepsilon_1(x^1_i - x^1_j)^2 + \cdots + \varepsilon_n(x^n_i - x^n_j)^2 + \varepsilon(x^{n+1}_i - x^{n+1}_j)^2]e^{2w_i+2w_j}. $$
The embedding method is also applicable to other geometries of local maximum mobility, which gives us the hope of constructing a complete classification of such geometries.