  We prove that the Cauchy problem for parallel null vector fields on smooth
Lorentzian manifolds is well posed. The proof is based on the derivation and
analysis of suitable hyperbolic evolution equations given in terms of the Ricci
tensor and other geometric objects. Moreover, we classify Riemannian manifolds
satisfying the constraint conditions for this Cauchy problem. It is then
possible to characterise certain holonomy reductions of globally hyperbolic
manifolds with parallel null vector in terms of flow equations for Riemannian
special holonomy metrics. For exceptional holonomy groups these flow equations
have been investigated in the literature before in other contexts. As an
application, the results provide a classification of Riemannian manifolds
admitting imaginary generalised Killing spinors. We will also give new local
normal forms for Lorentzian metrics with parallel null spinor in any dimension.
