  A local classification of locally conformal flat Riemannian Einstein-like
four-manifolds as well as a local classification of all locally conformal flat
Riemannian four-manifolds for which all Jacobi operators have parallel
eigenspaces along every geodesic is given. Non-trivial explicit examples are
presented. The problem of local description of self-dual Einstein-like
four-manifolds is also treated. A complete explicit solution of the St\"ackel
system in dimension four is obtained.
