  Systematic numerical investigations of the asymptotics of near Schwarzschild
vacuum initial data sets is carried out by inspecting solutions to the
parabolic-hyperbolic and to the algebraic-hyperbolic forms of the constraints,
respectively. One of our most important findings is that the concept of near
Schwarzschild configurations, applied previously in [4, 5], is far too
restrictive. It is demonstrated that by relaxing the conditions on the freely
specifiable part of the data a more appropriate notion of near Schwarzschild
initial data configurations can be defined which allows us to generate
asymptotically flat initial data configurations.
