  We prove that given a pseudo-Riemannian conformal structure whose conformal
holonomy representation fixes a totally lightlike subspace of arbitrary
dimension, there is, wrt. a local metric in the conformal class defined off a
singular set, a parallel, totally lightlike distribution on the tangent bundle
which contains the image of the Ricci-tensor. This generalizes results obtained
for invariant lightlike lines and planes and closes a gap in the understanding
of the geometric meaning of reducibly acting conformal holonomy groups. We show
how this result naturally applies to the classification of geometries admitting
twistor spinors in some low-dimensional split signatures when they are
described using conformal spin tractor calculus. Together with already known
results about generic distributions in dimensions 5 and 6 we obtain a complete
geometric description of local geometries admitting real twistor spinors in
signatures (3,2) and (3,3). In contrast to the generic case where generic
geometric distributions play an important role, the underlying geometries in
the non-generic case without zeroes turn out to admit integrable distributions.
