  We show that in the analytic category, given a Riemannian metric $g$ on a
hypersurface $M\subset \Z$ and a symmetric tensor $W$ on $M$, the metric $g$
can be locally extended to a Riemannian Einstein metric on $Z$ with second
fundamental form $W$, provided that $g$ and $W$ satisfy the constraints on $M$
imposed by the contracted Codazzi equations. We use this fact to study the
Cauchy problem for metrics with parallel spinors in the real analytic category
and give an affirmative answer to a question raised in B\"ar, Gauduchon,
Moroianu (2005). We also answer negatively the corresponding questions in the
smooth category.
