  We characterize certain CR structures of arbitrary codimension (different
from 3, 4 and 5) on Riemannian Spin$^c$ manifolds by the existence of a
Spin$^c$ structure carrying a strictly partially pure spinor field.
Furthermore, we study the geometry of Riemannian Spin$^c$ manifolds carrying a
strictly partially pure spinor which satisfies the generalized Killing equation
in prescribed directions.
