  Lorentzian manifolds with parallel spinors are important objects of study in
several branches of geometry, analysis and mathematical physics. Their Cauchy
problem has recently been discussed by Baum, Leistner and Lischewski, who
proved that the problem locally has a unique solution up to diffeomorphisms,
provided that the intial data given on a space-like hypersurface satisfy some
constraint equations. In this article we provide a method to solve these
constraint equations. In particular, any curve (resp. closed curve) in the
moduli space of Riemannian metrics on $M$ with a parallel spinor gives rise to
a solution of the constraint equations on $M\times (a,b)$ (resp. $M\times
S^1$).
