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Can Eyeball Planets Be Tidally Locked?

Intro
It's not at all clear to me that an eyeball planet is possible. By definition eyeball planets are tidally locked which implies that the rocky planet, any surface water or ice as well as it's atmosphere all remain in a fairly stable distribution around the planet. Because the planet is tidally locked it would result in looking like an "eye".

There are two scenarios for which I am curious to know more about: a) planets with tremendous amounts of water and b) planets with modest amounts of water.

Large Water Tidal Lock
This hypothetical planet would have a tremendous ocean on one side of the planet, ice on the other, and something slushy around the belt.

I suppose this is possible but I strongly suspect that the awkward distribution of mass and/or density of the water and ice would force the planet to tilt and would break the planet free from tidal lock and force a rotation.

Alternatively it could be possible to be tidally locked but tremendous ocean currents (from same effect of unbalanced or wobbly planet) would redistribute the water, probably with a small ice cap on the other end. This "dark side" warming would be assisted by a green house type of atmosphere.

Small Water Tidal Lock
In this case I would not expect tidal lock to break. However the planet would still be unbalanced and unstable. I suspect that these planets would be subject to rare, but regular occurrences of "flipping" where the entire planet leaves its metastable stable, massive surface redistribution of ice and water before settling again, temporarily, into another tidally locked position.

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