  In this paper, we consider a compact Riemannian manifold whose boundary is
endowed with a Riemannian flow. Under a suitable curvature assumption depending
on the O'Neill tensor of the flow, we prove that any solution of the basic
Dirac equation is the restriction of a parallel spinor field defined on the
whole manifold. As a consequence, we show that the flow is a local product. In
particular, in the case where solutions of the basic Dirac equation are given
by basic Killing spinors, we characterize the geometry of the manifold and the
flow.
