  We reexamine from first principles the classical Goldberg-Sachs theorem from
General Relativity. We cast it into the form valid for complex metrics, as well
as real metrics of any signature. We obtain the sharpest conditions on the
derivatives of the curvature that are sufficient for the implication
(integrability of a field of alpha planes)$\Rightarrow$(algebraic degeneracy of
the Weyl tensor). With every integrable field of alpha planes we associate a
natural connection, in terms of which these conditions have a very simple form.
