  This article uses the conformal Einstein equations and the conformal
representation of spatial infinity introduced by Friedrich to analyse the
behaviour of the gravitational field near null and spatial infinity for the
development of initial data which are, in principle, non-conformally flat and
time asymmetric. This article is the continuation of the investigation started
in Class. Quantum Grav. 21 (2004) 5457-5492, where only conformally flat
initial data sets were considered. For the purposes of this investigation, the
conformal metric of the initial hypersurface is assumed to have a very
particular type of non-smoothness at infinity in order to allow for the
presence of non-Schwarzschildean initial data sets in the class under study.
The calculation of asymptotic expansions of the development of these initial
data sets reveals --as in the conformally flat case-- the existence of a
hierarchy of obstructions to the smoothness of null infinity which are
expressible in terms of the initial data. This allows for the possibility of
having spacetimes where future and past null infinity have different degrees of
smoothness. A conjecture regarding the general structure of the hierarchy of
obstructions is presented.
