The momentum constraints of general relativity and spatial conformal isometries

Transverse-tracefree (TT-) tensors on (R(3),g(ab)), With g(ab) an asymptotically flat metric of fast decay at infinity, are studied. When the source tensor from which these TT tensors are constructed has fast fall-off at infinity, TT tensors allow a multipole-type expansion. When g(ab) has no conformal Killing vectors (CKV's) it is proven that any finite but otherwise arbitrary set of moments can be realized by a suitable TT tenser. When CKV's exist there are obstructions-certain (combinations of) moments have to vanish-which we study.