On a family of functional equations

The $(n+1)$-dimensional geometry of local maximum mobility is given by some non-degenerate and differentiable function $f$ of the pair of points on the manifold $M$, which is а motion group invariant of dimension $(n+1)(n+2)/2$. There is no complete classification of such geometries of dimension $n + 1$, but some examples are well known: Euclidean geometry, symplectic geometry, constant curvature geometry. Recently, some previously unknown geometries of local maximum mobility has been found using the embedding method. The embedding method enables one to find functions $f$ that define $(n+1)$-dimensional geometries of local maximum mobility by functions $\theta$ of known $n$-dimensional geometry of local maximum mobility. This problem is reduced to the solution of functional equations of a special type, which are a consequence of the invariance of the function $f$ of the pair of points with respect to the motion group. Such equations are solved in this paper. By differentiation, they are first reduced to functional differential equations, from which we pass to differential equations by separating the variables. Then the solutions of the latter are substituted into the original functional equations, after which we get the final result.