  We study the Einstein-Dirac equation as well as the weak Killing equation on
Riemannian spin manifolds with codimension one foliation. We prove that, for
any manifold $M^n$ admitting real Killing spinors (resp. parallel spinors),
there exist warped product metrics $\bar{\eta}$ on $M^n \times {\mathbb R}$
such that $(M^n \times {\mathbb R}, \bar{\eta})$ admit Einstein spinors (resp.
weak Killing spinors). To prove the result we split the Einstein-Dirac equation
into evolution equations and constraints, by means of Cartan's frame formalism,
and apply the local preservation property of constraints.
