Pseudo-Riemannian manifolds with recurrent spinor fields

The existence of a recurrent spinor field on a pseudo-Riemannian spin manifold ($M,g$) is closely related to the existence of a parallel $1$-dimensional complex subbundle of the spinor bundle of ($M,g$). We characterize the following simply connected pseudo-Riemannian manifolds that admit these subbundles in terms of their holonomy algebras: Riemannian manifolds, Lorentzian manifolds, pseudo-Riemannian manifolds with irreducible holonomy algebras, and pseudo-Riemannian manifolds of neutral signature admitting two complementary parallel isotropic distributions.