In this paper we prove density of asymptotically flat solutions with
special asymptotics in general classes of solutions of the vacuum constraint
equations. The first type of special asymptotic form we consider is called harmonic asymptotics.
This generalizes in a natural way the conformally flat asymptotics for the K = 0 constraint equations.
We show that solutions with harmonic asymptotics form a dense subset (in a suitable weighted
Sobolev topology) of the full set of solutions. An important feature of this construction is that
the approximation allows large changes in the angular momentum. The second density theorem we prove allows us to approximate
asymptotically flat initial data on a three-manifold M for the vacuum Einstein field equation by solutions which agree with
the original data inside a given domain, and are identical to that of a suitable Kerr slice (or identical
to a member of some other admissible family of solutions) outside a large ball in a given end. The
construction generalizes work in [C], where the time-symmetric (K = 0) case was studied.