We show that in the analytic category, given a Riemannian metric g on a
hypersurfaceM � Z and a symmetric tensorW onM, 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 a�rmative answer to a question raised in [15]. We also
answer negatively the corresponding questions in the smooth category.