  To solve the problem of exact integration of the field equations or equations
of motion of matter in curved spacetimes one can use a class of Riemannian
metrics for which the simplest equations of motion can be integrated by the
complete separation of variables method. Here, we consider the particular case
of the class of Stackel metrics. These metrics admit integration of the
Hamilton-Jacobi equation for test particle by the complete separation of
variables method. It appears that the other important equations of motion
(Klein-Gordon-Fock, Dirac, Weyl) in curved spacetimes can be integrated by
complete separation of variables method only for the metrics, belonging to the
class of Stackel spaces.
