  The complete solution of Einstein's gravitational equations with a
vacuum-vacuum Kerr-Schild pencil of metrics $g_{ab}+V l_al_b$ is obtained. Our
result generalizes the solution of the Kerr-Schild problem with a flat metric
$g_{ab}$ (represented by the Kerr theorem) to the case when $g_{ab}$ is the
metric of a curved space-time.
