What is meant by the bifurcation of hydrostatic equilibrium shapes in planetary formation?
I think all this means is that up to some critical rotation rate, the equilibrium shape of a rotating, self-gravitating fluid is an oblate spheroid. That is, this shape defines a unique global minimum of energy.

Above this threshold there are two possible equilibrium solutions. Apparently one is an oblate spheroid, whereas the other is a triaxial ellipsoid. These will define local energy minima, but the triaxial ellipsoid is the global minimum.

The gory mathematical details are no doubt contained in Chandrasekhar's textbook on equilibrium ellipsoidal figures. Pages 3-5 of Iurato (2014) appear to summarise the historical development of this bifurcation in the equilibrium solutions.