  A theorem providing a characterisation of Schwarzschildean initial data sets
on slices with an asymptotically Euclidean end is proved. This characterisation
is based on the proportionality of the Weyl tensor and its D'Alambertian that
holds for some vacuum Petrov Type D spacetimes (e.g. the Schwarzschild
spacetime, the C-metric, but not the Kerr solution). The 3+1 decomposition of
this proportionality condition renders necessary conditions for an initial data
set to be a Schwarzschildean initial set. These conditions can be written as
quadratic expressions of the electric and magnetic parts of the Weyl tensor
--and thus, involve only the freely specifiable data. In order to complete our
characterisation, a study of which vacuum static Petrov type D spacetimes admit
asymptotically Euclidean slices is undertaken. Furthermore, a discussion of the
ADM 4-momentum for boost-rotation symmetric spacetimes is given. Finally, a
generalisation of our characterisation, valid for Schwarzschildean
hyperboloidal initial data sets is put forward.
