http://mathoverflow.net/questions/208901/separable-coordinate-systems-for-the-laplace-and-helmholtz-equations

What I want to show you in particular are these papers:

- R-Separation for heat and Schrodinger equations I (page 661)

Separation of variables for complex riemannian spaces of constant curvature. I. (end of page 201)

The papers are fairly intractable, and a lot of the symmetry notation is unfamiliar to me. But in both papers they are talking about coordinate systems based on rotatopes. Specifically, they have a list of coordinate systems that includes all the rotatope-based coordinate systems, but also includes parabolic and elliptic-type coordinates. They categorise them based on partitions of n, just like us. Personally I think the notation they use is horrible though. I haven't had time to read them thoroughly, but I think they also talk about cyclidal coordinate systems, which correspond to the quartic toratopes.