  We present an analysis of the vacuum Einstein equations for a recently
proposed extension of the Kerr-Schild ansatz that includes a spacelike vector
field as well as the usual Kerr-Schild null vector. We show that many, although
not all, of the simplifications that occur in the Kerr-Schild case continue to
hold for the extended ansatz. In particular, we find a simple set of sufficient
conditions on the vectors such that the vacuum field equations truncate beyond
quadratic order in an expansion around a general vacuum background solution. We
extend our analysis to the electrovac case with a related ansatz for the gauge
field.
