Before I continue to spend time on this, let me make sure that the following
is OK. As Mallinckrodt I can obviously only do the task for a case that
would correspond to circular orbits in inertial frame. That's why I asked
the question originally. There never "in general" exists a frame in which
two independent bodies in arbitrary motion remain at rest. Only for
specific particular cases can that be; and I'm assuming that it is OK if I
choose a situation that corresponds to such a particular case; namely the
frame specified, which correspondes to circular orbits.
I hope this is satisfactory to everyone, if not let me know.