Summary: It is shown that if a space-time V admits a covariantly constant spinor field \(\psi_ A\), and hence a covariantly constant null vector field \(\ell_{\mu}\) determined by \(\psi_ A\), its Ricci tensor is proportional to the tensor product of \(\ell_{\mu}\) by itself. Further, the conformal tensor of V is shown to be of Petrov-Penrose type N. That is the four index symmetric spinor determined by the conformal tensor of V is proportional to the spinor product of \(\psi_ A\) with itself. The Bianchi identities are used to show that empty asymptotically flat space- times with infinitely extendible null geodesics tangent to \(\ell_{\mu}\) are flat.