A Partial Differential Equation which can be written in a Scalar version
(1) 
(2) 
The Helmholtz differential equation can be solved by Separation of Variables in only 11 coordinate systems, 10 of which (with the exception of Confocal Paraboloidal Coordinates) are particular cases of the Confocal Ellipsoidal system: Cartesian, Confocal Ellipsoidal, Confocal Paraboloidal, Conical, Cylindrical, Elliptic Cylindrical, Oblate Spheroidal, Paraboloidal, Parabolic Cylindrical, Prolate Spheroidal, and Spherical Coordinates (Eisenhart 1934). Laplace's Equation (the Helmholtz differential equation with ) is separable in the two additional Bispherical Coordinates and Toroidal Coordinates.
If Helmholtz's equation is separable in a 3D coordinate system, then Morse and Feshbach (1953, pp. 509510) show that
(3) 

(4) 
(5) 
(6) 
See also Laplace's Equation, Poisson's Equation, Separation of Variables, Spherical Bessel Differential Equation
References
Eisenhart, L. P. ``Separable Systems in Euclidean 3Space.'' Physical Review 45, 427428, 1934.
Eisenhart, L. P. ``Separable Systems of Stäckel.'' Ann. Math. 35, 284305, 1934.
Eisenhart, L. P. ``Potentials for Which Schroedinger Equations Are Separable.'' Phys. Rev. 74, 8789, 1948.
Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGrawHill, pp. 125126 and 509510, 1953.
© 19969 Eric W. Weisstein