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. I suspect it remains unsolved. That is: There perhaps also exist non-ellipsoidal solutions (which remain unknown), and indeed in some regimes those might be the stable solutions. I would suspect a "spinning solid torus" solution probably exists, for example, albeit I doubt it could be stable. I don't think unstability really matters, in the sense that, if you somehow constructed such a planet, then froze the liquid, the result then might survive because the strength of the solid would prevent the small perturbations which would for a liquid yield instability. Anybody who wants to know would be advised to begin by reading S.Chandrasekhar's book (I haven't).