RR-separation of variables for the conformally invariant Laplace–Beltrami equation

The conditions for RR-separation of variables for the conformally invariant Laplace–Beltrami equation on an nn-dimensional pseudo-Riemannian manifold are determined and compared with the conditions for the additive separation of the null geodesic Hamilton–Jacobi equation. The case of three-dimensions is examined in detail and it is proven that on any conformally flat manifold the two equations separate in the same coordinates.