We classify Riemannian spinᶜ manifolds carrying a type I imaginary generalized Killing
spinor, by explicitly constructing a parallel spinor on each leaf of the canonical foliation given by the Dirac current. We also provide a class of Riemannian spinᶜ manifolds carrying a type II imaginary generalized Killing spinor, by considering spacelike hypersurfaces of Lorentzian spinᶜ manifolds.We carry out much of thework in the setting of semi-Riemannian spinᶜ-manifolds carrying generalized Killing spinors, allowing us to draw conclusions in this setting as well. In this context, the Dirac current is not always a closed vector field. We circumvent this in even dimensions, by considering a modified Dirac current, which is closed in the cases when the original Dirac current is not. On the path to these results, we also study
semi-Riemannian manifolds carrying closed and conformal vector fields.