  Non-conformally flat space-times admitting a non-null Killing spinor of
valence two are investigated in the Geroch-Held-Penrose formalism. Contrary to
popular belief these space-times are not all explicitly known. It is shown that
the standard construction hinges on the tacit assumption that certain
integrability conditions hold, implying two algebraic relations, KS1 and KS2,
for the spin coefficients and the components of the Ricci spinor. An exhaustive
list of (conformal classes of) space-times, in which either KS1 or KS2 are
violated, is presented. The resulting space-times are each other's Sachs
transforms, in general admit no Killing vectors and are characterized by a
single arbitrary function.
