Separations of Variables and Analytic Contractions on Two-Dimensional Hyperboloids

In this review we present recent results in the field of analytical contraction of Lie algebra in two-dimensional hyperbolic space. A complete geometric description for all possible orthogonal and nonorthogonal (related to the first order symmetries) systems of coordinates, which allow separation of variables of two-dimensional Laplace–Beltrami or Helmholtz equation on the two-sheeted (upper sheet) $${{H}_{2}}$$ and the one-sheeted $${{\tilde {H}}_{2}}$$ hyperboloids is given. The limiting transition between non subgroup (mostly parametric) and subgroup systems is conducted. The analytic contractions between various systems of coordinates in two hyperbolic spaces and Euclidean $${{E}_{2}}$$ and Minkowski $${{E}_{{1,1}}}$$ spaces are presented.