I did a similar search a few years ago, and uncovered discussions of fractal behavior when the rotation spins too fast. The behavior is obviously different with surface-tension/self-gravitational/self-GR liquid blobs. The problem is in keeping rotational symmetry which becomes essentially impossible after a certain point -- the whole configuration blows apart into smaller blobs. I seem to recall that the shape just prior to blowing apart looks somewhat like an Olympic discus, with a thicker part in the center, but an inflection point part way out on the rim. At 07:33 PM 3/5/2015, Warren D Smith wrote:

http://people.ucsc.edu/~igarrick/EART290/chandrasekhar_1967.pdf

is a paper by S.Chandrasekhar about the history of the theory of self-gravitating fixed-shape blobs of constant density fluid. Truly, the architects of this theory were The Masters.

Anyhow, their main result is that certain ellipsoids are exact solutions. I wonder whether any "Jacobi ellipsoids" actually occur in Nature as asteroids or comets -- ellipsoids with 3 unequal axes, in perfect hydrostatic equilibrium.

I do not know if, to this day, it has been proven that these are the only solutions.