Eigenfunctions of the Beltrami–Laplace operator on a hyperboloid of one sheet

The Beltrami–Laplace operator $\Delta$ on a one-sheeted hyperboloid is hyperbolic. There is taken the set of functions bounded outside some neighborhood of two isotropic lines, intersecting at infinity. A necessary and sufficient condition is derived that a function of this set be an eigenfunction of the operator $\Delta$.