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Is it true that all static solutions in GR are also spherically symmetric? Is there a proof of this?

Similarly, are all stationary solutions axisymmetric?

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The answer to your first question is no. In fact, you can find static, stationary solutions to GR corresponding to cosmic strings and domain walls, or even more exotic solutions, like the c-metric.

Your statement about axisymmetry is harder, mainly because I don't know of many non-axisymmetric solutions, period.

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