Consider the Laplace-Beltrami operator on a compact manifold. Generically, Uhlenbeck has shown that eigenfunctions of the Laplace-Beltrami operator are Morse functions.
But there are some manifolds, such as the two-dimensional torus, where some eigenfunctions of the Laplace-Beltrami operator have non-isolated maxima.
For the only examples I know of, the eigenfunctions are separable, i.e. they can be written as a product of lower-dimensional functions. I heard somewhere that separable eigenfunctions are the only eigenfunctions whose critical points are not isolated, but I can't remember where to find it, or even where to look.
Question. Is this true, and does anyone know of a reference or how to prove it?