In Einstein-Maxwell gravity with a conformally coupled scalar field, the black hole found by Bocharova, Bronnikov, Melnikov, and Bekenstein breaks when embedded in the external magnetic field of the Melvin universe. The situation improves in presence of acceleration, allowing one to build magnetised and accelerating BBMB black hole with a thin membrane. But to overcome this and others disadvantages of BBMB spacetimes, a new class of black holes, including the rotating case, is proposed for the conformal matter coupling under consideration.
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