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Kerr black hole immersed in test, asymptotically homogeneous magnetic field, aligned along the symmetry axis, is described by Wald's solution. We show how this solution may be generalized for nonlinear electromagnetic models via perturbative approach. Using this technique we find the lowest order correction to Wald's solution on Schwarzschild spacetime in Euler-Heisenberg and Born-Infeld models. Finally, we discuss the problem of highly conducting star in asymptotically homogeneous magnetic field.

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Discovery that gravitational field equations may coerce the spacetime metric with isometries to attain a block-diagonal form compatible with these isometries, was one of the gems built into the corpus of black hole uniqueness theorems. We revisit the geometric background of a block-diagonal metric with isometries, foliation defined by Killing vector fields and the corresponding Godbillon--Vey characteristic class. Furthermore, we analyse sufficient conditions for various matter sources, including scalar, nonlinear electromagnetic and Proca fields, that imply the isometry-compatible block-diagonal form of the metric. Finally, we generalize the theorem on the absence of null electromagnetic fields in static spacetimes to an arbitrary number of spacetime dimensions, wide class of gravitational field equations and nonlinear electromagnetic fields.