Subgroup type coordinates and the separation of variables in   Hamilton-Jacobi and Schr\H{o}dinger equations
  Separable coordinate systems are introduced in the complex and real four-dimensional flat spaces. We use maximal Abelian subgroups to generate coordinate systems with a maximal number of ignorable variables. The results are presented (also graphically) in terms of subgroup chains. Finally, the explicit solutions of the Schr\H{o}dinger equation in the separable coordinate systems are computed. 
E.G. Kalnins, Z. Thomova and P. Winternitz