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Transverse-tracefree (TT-) tensors on $({\bf R}^3,g_{ab})$, with $g_{ab}$ anasymptotically flat metric of fast decay at infinity, are studied. When thesource tensor from which these TT tensors are constructed has fast fall-off atinfinity, TT tensors allow a multipole-type expansion. When $g_{ab}$ has noconformal Killing vectors (CKV's) it is proven that any finite but otherwisearbitrary set of moments can be realized by a suitable TT tensor. When CKV'sexist there are obstructions -- certain (combinations of) moments have tovanish -- which we study.Comment: 16 page

The momentum constraints of general relativity and spatial conformal isometries