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This article uses the conformal Einstein equations and the conformalrepresentation of spatial infinity introduced by Friedrich to analyse thebehaviour of the gravitational field near null and spatial infinity for thedevelopment of initial data which are, in principle, non-conformally flat andtime asymmetric. This article is the continuation of the investigation startedin Class. Quantum Grav. 21 (2004) 5457-5492, where only conformally flatinitial data sets were considered. For the purposes of this investigation, theconformal metric of the initial hypersurface is assumed to have a veryparticular type of non-smoothness at infinity in order to allow for thepresence of non-Schwarzschildean initial data sets in the class under study.The calculation of asymptotic expansions of the development of these initialdata sets reveals --as in the conformally flat case-- the existence of ahierarchy of obstructions to the smoothness of null infinity which areexpressible in terms of the initial data. This allows for the possibility ofhaving spacetimes where future and past null infinity have different degrees ofsmoothness. A conjecture regarding the general structure of the hierarchy ofobstructions is presented.Comment: Second part of gr-qc/0408062, 23 page

Time asymmetric spacetimes near null and spatial infinity: II. Expansions of developments of initial data sets with non-smooth conformal metrics