  We consider weighted parallel spinors in Lorentzian Weyl geometry in
arbitrary dimensions, choosing the weight such that the integrability condition
for the existence of such a spinor, implies the geometry to be Einstein-Weyl.
We then use techniques developed for the classification of supersymmetric
solutions to supergravity theories to characterise those Lorentzian EW
geometries that allow for a weighted parallel spinor, calling the resulting
geometries supersymmetric. The overall result is that they are either
conformally related to ordinary geometries admitting parallel spinors (w.r.t.
the Levi-Civita connection) or are conformally related to certain Kundt
spacetimes. A full characterisation is obtained for the 4 and 6 dimensional
cases.
