  The Goldberg-Sachs theorem is generalized for all four-dimensional manifolds
endowed with torsion-free connection compatible with the metric, the treatment
includes all signatures as well as complex manifolds. It is shown that when the
Weyl tensor is algebraically special severe geometric restrictions are imposed.
In particular it is demonstrated that the simple self-dual eigenbivectors of
the Weyl tensor generate integrable isotropic planes. Another result obtained
here is that if the self-dual part of the Weyl tensor vanishes in a Ricci-flat
manifold of (2,2) signature the manifold must be Calabi-Yau or symplectic and
admits a solution for the source-free Einstein-Maxwell equations.
