  Generic distributions on 5- and 6-manifolds give rise to conformal structures
that were discovered by P. Nurowski resp. R. Bryant. We describe both as
Fefferman-type constructions and show that for orientable distributions one
obtains conformal spin structures. The resulting conformal spin geometries are
then characterized by their conformal holonomy and equivalently by the
existence of a twistor spinor which satisfies a genericity condition. Moreover,
we show that given such a twistor spinor we can decompose a conformal Killing
field of the structure. We obtain explicit formulas relating conformal Killing
fields, almost Einstein structures and twistor spinors.
