On pp-waves with lightlike parallel spinors

We parametrize pp-wave spacetimes with compact codimension 2 hypersurfaces. In the vacuum case, we show that these spacetimes are locally in one-to-one correspondence with smooth curves of Riemannian Ricci-flat metrics modulo smooth curves of diffeomorphisms. We also prove that this one-to-one correspondence extends to pp-waves with prescribed null Ricci curvature. Moreover, the pp-wave spacetime carries a lightlike parallel spinor if and only if one (and hence all) of the Ricci-flat metrics carries a parallel spinor.