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The Levi-Civita functional equation $f\left(gh\right)={\sum }_{k=1}^n{u}_k\left(g\right)v_k\left(h\right)$ (g,h ∈ G), for scalar functions on a topological semigroup G, has as the solutions the functions which have finite-dimensional orbits in the right regular representation of G, that is the matrix elements of G. In considerations of some extensions of the L-C equation one encounters with other geometric problems, for example: 1) which vectors x of the space X of a representation $g\mapsto T_g$ have orbits O(x) that are “close” to a fixed finite-dimensional subspace? 2) for...