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The equilibrium of incompressible spheroid-ring systems in rigid rotation isinvestigated by numerical means for a unity density contrast. A great diversityof binary configurations is obtained, with no limit neither in the mass ratio,nor in the orbital separation. We found only detached binaries, meaning thatthe end-point of the $\epsilon_2$-sequence is the single binary state in strictcontact, easily prone to mass-exchange. The solutions show a remarkableconfinement in the rotation frequency-angular momentum diagram, with a totalabsence of equilibrium for $\Omega^2/ \pi G \rho \gtrsim 0.21$. A short band ofdegeneracy is present next to the one-ring sequence. We unveil a continuum ofbifurcations all along the ascending side of the Maclaurin sequence foreccentricities of the ellipsoid less than $\approx 0.612$ and which involves agradually expanding, initially massless loop.

Rigidly rotating, incompressible spheroid-ring systems: new bifurcations, critical rotations and degenerate states