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