  Off-shell formulations of supergravities allow one to add closed-form
higher-derivative super-invariants that are separately supersymmetric to the
usual lower-derivative actions. In this paper we study four-dimensional
off-shell N=1 supergravity where additional super-invariants associated with
the square of the Weyl tensor and the square of the Ricci scalar are included.
We obtain a variety of solutions where the metric describes domain walls,
Lifshitz geometries, and also solutions of a kind known as gyratons. We find
that in some cases the solutions can be supersymmetric for appropriate choices
of the parameters. In some solutions the auxiliary fields may be imaginary. One
may reinterpret these as real solutions in an analytically-continued theory.
Since the supersymmetry transformation rules now require the gravitino to be
complex, the analytically-continued theory has a "fake supersymmetry" rather
than a genuine supersymmetry. Nevertheless, the concept of
pseudo-supersymmetric solutions is a useful one, since the Killing spinor
equations provide first-order equations for the bosonic fields.
