Is it possible that two toroidal, rigid, homogeneous planets are forming a system in which the toroids are chained through each other, and during their mutual motion, the planets are never touching each other? How should I go about studying the systems of the above kind, and classify them according to their geometrical parameters, masses and initial dynamical parameters.

Are there systems which are stable? (I considered the fact that there are a sizeable number of dynamical degrees of freedom.) What happens in the case for example, if the ‘orbit’ frequency reduces a bit, or if the plane of one of the toroids tilts slightly?

(I'm aware that Analytic calculations are favorable. For numerical calculations, the results should be cross-checked in some semi-analytic way, what would that be?)