  A 3+1 decomposition of the twistor and valence-2 Killing spinor equation is
made using the space spinor formalism. Conditions on initial data sets for the
Einstein vacuum equations are given so that their developments contain
solutions to the twistor and/or Killing equations. These lead to the notions of
twistor and Killing spinor initial data. These notions are used to obtain a
characterisation of initial data sets whose development are of Petrov type N or
D.
