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Previously the DeTurck 'trick' has been used to render the stationary Einstein's equation a well posed elliptic system that may be solved numerically by geometric flow or directly.
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P. Figueras, J. Lucietti, and T. Wiseman, Class. Quant. Grav. 28
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B. P. Abbott et al. (Virgo, LIGO Scientific), Phys. Rev. Lett. 116
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The Kerr spacetime is also algebraically special, namely Petrov type D
Cited in the paper.
In general, for vacuum spacetimes for D − 2 D-2 commuting Killing vector fields the Einstein’s equation is integrable, where D D is the total number of spacetime dimensions. Note that only in D = 4 , 5 D=4,5 these spacetimes can be asymptotically flat
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O. J. C. Dias, J. E. Santos, and B. Way, Class. Quant. Grav. 33
2016
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