  We prove that a smooth Riemannian manifold admitting an imaginary generalized
Killing spinor whose Dirac current satisfies an additional algebraic constraint
condition can be embedded as spacelike Cauchy hypersurface in a smooth
Lorentzian manifold on which the given spinor extends to a null parallel
spinor. This is in contrast to a corresponding Cauchy problem for real
generalized Killing spinors into Riemannian manifolds. The construction is
based on first order symmetric hyperbolic PDE-methods. In fact, the coupled
evolution equations for metric and spinor as considered here extend and
generalize the well known PDE-system appearing in the Cauchy problem for the
vacuum Einstein equations. Special cases are discussed and the statement is
compared with a similar result obtained recently for the analytic category.
