  Solutions of five-dimensional De Sitter supergravity admitting Killing
spinors are considered, using spinorial geometry techniques. It is shown that
the "null" solutions are defined in terms of a one parameter family of
3-dimensional constrained Einstein-Weyl spaces called Gauduchon-Tod structures.
They admit a geodesic, expansion-free, twist-free and shear-free null vector
field and therefore are a particular type of Kundt geometry. When the
Gauduchon-Tod structure reduces to the 3-sphere, the null vector becomes
recurrent, and therefore the holonomy is contained in Sim(3), the maximal
proper subgroup of the Lorentz group SO(4,1). For these geometries, all scalar
invariants built from the curvature are constant. Explicit examples are
discussed.
