Variable-separation theory for the null Hamilton–Jacobi equation

The theory of the separation of variables for the null Hamilton–Jacobi equation H=0 is systematically revisited and based on Levi–Civita separability conditions with Lagrangian multipliers. The separation of the null equation is shown to be equivalent to the ordinary separation of the image of the original Hamiltonian under a generalized Jacobi–Maupertuis transformation. The general results are applied to the special but fundamental case of the orthogonal separation of a natural Hamiltonian with a fixed value of the energy. The separation is then related to conditions which extend those of Stäckel and Kalnins and Miller (for the null geodesic case) and it is characterized by the existence of conformal Killing two-tensors of special kind.