The analytic method of embedding symplectic geometry

As you know, the $n$-dimensional geometry of maximum mobility allows the group of motions of dimension $n(n+1)/2$. Many of these geometries are well known such as euclidean and pseudoeuclidean geometries. These are phenomenologically symmetric geometries, i.e. for them the metric properties are equivalent to group ones. In this work we applied the analytical method of embedding, which helps to find metric functions of all three-dimensional geometries of maximum mobility, which contain as an argument metric functions of two-dimensional symplectic geometry.