I've read a lot more about separating PDEs. Things are actually simpler than they seem in a way, and much more complicated in another way.

There are two equations I'm interested in: The Helmholtz equation and Laplace's equation. Laplace's equation is simpler, which means there are more nice solutions available. Some coordinate systems are special in that these equations can be "separated" in them.

In 3D, the Helmholtz equation is separable in 11 different coordinate systems. But it turns out that they're all degenerate forms of ellipsoidal coordinates. These are related to the quadrics, but they don't map one to one with them. In fact it actually seems a bit subjective what you count as a "different" coordinate system.

The Laplace equation is separable in all of these, but there's also a different form of solution that permits two new coordinate systems: toroidal and bispherical. However these are also special cases of "cyclidal coordinates". In fact, I just realised that bispherical coordinates look just like toroidal coordinates, except that the coordinate surfaces are those dodgy torii that intersect themselves. I can hardly find anything about cyclides anywhere, but they're basically a generalization of the torus. The equation reminds me of the expanded equation of the torus. So they're a special type of quartic surface.

The method they use (at least in the old textbook I'm reading) for Laplace's equation is bizarre. It involves something called "pentaspherical coordinates" and a 5D version of Laplace's equation.

Now here's my conjectures for 4D:

1. The Helmholtz equation will be separable in hyperellipsoidal coordinates and all its degenerate forms, but nothing else.

2. Laplace's equation will be separable in all of these, plus a large number of 4D cyclidal coordinate systems, which will relate to toratopes in some way.

3. Possibly it will also be separable in some coordinate systems corresponding to octic (eighth order) surfaces, and maybe others. I think the toratopes ((II)(II)) and (((II)I)I) are both octic surfaces.