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.