The trajectories of two point masses or spherically symmetric masses with respect to their center of mass are conic sections or Kepler orbits.
Consider that the bodies have finite size with respect to their separation and not necessarily uniform, or even spherically symmetric mass distributions.
In that case what are the constraints on their mass distributions and orientations such that their orbits are still Keplerian? Or does any deviation from spherical symmetry of one or both body immediately result in a non-Keplerian orbit?
This answer to the Astronomy SE question Which point in an orbiting body most closely follows its Keplerian trajectory? explains:
If one or both of the bodies have non-spherically symmetric density distribution, the orbits will no longer be keplerian.
but without citing sources nor using math. I can certainly imagine "... are no longer necessarily Keplerian" but is this always true in absolute?
Just for example couldn't two tidally locked ellipsoids have circular orbits about their center of mass?
If so, perhaps that's the only exception, but perhaps not. If they orbited in non-circular orbits would their apparent libration also mean that their centers of mass no longer followed strict Keplerian orbits? Would they differ from ellipses, or still be elliptical but no longer exactly follow equal area per unit time trajectories? Would there be another point within the bodies besides their centers of mass that still did follow a Keplerian orbit?
Question: Which mass distributions guarantee that two bodies will always have non-Keplerian orbits? Which non-spherical distributions still allow for noncircular Keplerian orbits?