Field redefinitions, Weyl invariance and the nature of mavericks

In 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 potentials bounded from below). Probably the most famous example is the Bocharova-Bronnikov-Melnikov-Bekenstein (BBMB) black hole solution in a theory with a scalar field conformally coupled to the gravity, and with a vanishing potential. Its existence naively violates the 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, the nature of the BBMB solution is still the subject of research and debate. We argue here that the key to understanding the 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, the 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 analyses.