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Field redefinitions, Weyl invariance, and nature of mavericks

In the theories of gravity with non-minimally coupled scalar fields there are "mavericks" -- unexpected solutions with odd properties, e.g., black holes with scalar hair in theories with scalar potential bounded from below. Probably the most famous example is Bocharova-Bronnikov-Melnikov-Bekenstein (BBMB) black hole solution in a theory with a scalar field conformally coupled to the gravity and with vanishing potential. Its existence naively violates no-hair conjecture without violating no-hair theorems because of the singular behavior of the scalar field at the horizon. Despite being discovered more than 40 years ago, nature of BBMB solution is still the subject of research and debate. We argue here that the key in understanding nature of maverick solutions is the proper choice of field redefinition schemes in which the solutions are regular. It appears that in such "regular" schemes mavericks have different physical interpretations, in particular they are not elementary but composite objects. For example, BBMB solution is not an extremal black hole but a collection of a wormhole and a naked singularity. In the process we show that Weyl-invariant formulation of gravity is a perfect tool for such analyzes.