Solution of the embedding problem for two-dimensional and three-dimensional geometries of local maximum mobility

In modern geometry, the study of the geometries of maximum mobility is of great importance. Some of these geometries are well studied (for example, the Euclidean and Lobachevsky geometries, pseudo-Euclidean, symplectic, spherical geometry, etc.), while others (for example, Helmholtz and pseudo-Helmholtz geometries) have not yet attracted active attention of researchers. There is still no complete classification of the geometries of maximum mobility. In this work, we present some results concerning the classification problem for two- and three-dimensional geometries of local maximum mobility. This problem is reduced to functional equations of a special form and is solved by the embedding method in the class of analytic functions.