  We show that a class of solutions of minimal supergravity in five dimensions
is given by lifts of three--dimensional Einstein--Weyl structures of hyper-CR
type. We characterise this class as most general near--horizon limits of
supersymmetric solutions to the five--dimensional theory. In particular, we
deduce that a compact spatial section of a horizon can only be a Berger sphere,
a product metric on $S^1\times S^2$ or a flat three-torus. We then consider the
problem of reconstructing all supersymmetric solutions from a given
near--horizon geometry. By exploiting the ellipticity of the linearised field
equations we demonstrate that the moduli space of transverse infinitesimal
deformations of a near--horizon geometry is finite--dimensional.
