  Starting from a bundle E over R, the dual of the first jet bundle, which is a
co-dimension 1 sub-bundle of the cotangent bundle of E, is the appropriate
manifold for the geometric description of time-dependent Hamiltonian systems.
Based on previous work, we recall properties of the complete lifts of a type
(1,1) tensor R on E to both of these manifolds. We discuss how an interplay
between these lifted tensors leads to the identification of related
distributions on both manifolds. The integrability of these distributions, a
coordinate free condition, is shown to produce exactly Forbat's conditions for
separability of the time-dependent Hamilton-Jacobi equation in appropriate
coordinates.
