  In this work, a subclass of the generalized Kerr-Schild class of spacetimes
is specified, with respect to which the Ricci tensor (regardless of the
position of indices) proves to be linear in the so-called profile function of
the geometry. Considering Colombeau's nonlinear theory of generalized
functions, this result is extended to apply to an associated class of
distributional Kerr-Schild geometries, and then used to formulate a variational
principle for these singular spacetimes. More specifically, it is shown in this
regard that a variation of a suitably regularized Einstein-Hilbert action can
be performed even if the metric of one of the corresponding generalized
Kerr-Schild representatives contains a generalized delta function that
converges in a suitable limit to a delta distribution.
