Ernst's solution-generating technique is adapted to Einstein-Maxwell theory conformally (and minimally) coupled to a scalar field. This integrable system enjoys an SU(2,1) symmetry which enables one to move, by Kinnersley transformations, through the axisymmetric and stationary solution space, building an infinite tower of physically inequivalent solutions. As a specific application, metrics associated to scalar hairy black holes---such as the ones discovered by Bocharova, Bronnikov, Melnikov, and Bekenstein---are embedded in the external magnetic field of the Melvin universe.
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Marco Astorino (2013) studied this question.
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