Complex variables for separation of Hamilton-Jacobi equation on real pseudo-Riemannian manifolds

In this paper the geometric theory of separation of variables for time-independent Hamilton-Jacobi equation is extended to include the case of complex eigenvalues of a Killing tensor on pseudo-Riemannian manifolds. This task is performed without to complexify the manifold but just considering complex-valued functions on it. The simple formalism introduced allows to extend in a very natural way the classical results on separation of variables (including Levi-Civita criterion and Stackel-Eisenhart theory) to the complex case. Orthogonal variables only are considered.