An exact and analytical solution, in four-dimensional general relativity coupled with Maxwell electromagnetism, is built by means of a Lie point symmetry of the Ernst equations, the Harrison transformation. The new spacetime describes a Schwarzschild-like black hole embedded into a general external backreacting electromagnetic field, which is the superposition of the Levi-Civita-Bertotti-Robinson and the Bonnor-Melvin ones. The relation between the two homogeneous electromagnetic fields is clarified. Conserved charges and the first law of thermodynamics are analyzed. Swirling generalizations are also considered. Limits to the known metrics such as Schwarzschild-Bertotti-Robinson, Schwarzschild-Bonnor-Melvin, and Bertotti-Robinson-Bonnor-Melvin are discussed.

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• Classical black holes
• General relativity
• Interplanetary magnetic field
• Integrable systems