In their Field Theory Handbook, while discussing eleven coordinate systems in which Helmholtz equation is separable, Moon and Spencer state that: "Skew coordinate systems do not allow separation of variables and will not be considered."
Since they do not provide any proof of this statement, I presume that this should be self-evident. However, I am unable to see why this must be true and would appreciate any clarification or, better yet, a rigorous proof.
My question: why don't skew coordinate systems allow separation of variables?
As far as I know, this is a nontrivial theorem.
For a similar statement in a related but slightly different setting, see Section 12 in the paper Separability in Riemannian Manifolds by Sergio Benenti (2016).