  We consider four dimensional stationary and axially symmetric spacetimes for
conformally coupled scalar-tensor theories. We show that, in analogy to the
Lewis-Papapetrou problem in General Relativity (GR), the theory at hand can be
recast in an analogous integrable form. We give the relevant rod formalism,
introduced by Weyl for vacuum GR, explicitly giving the rod structure of the
black hole of Bocharova et al. and Bekenstein (BBMB), in complete analogy to
the Schwarzschild solution. The additional scalar field is shown to play the
role of an extra Weyl potential. We then employ the Ernst method as a concrete
solution generating example to obtain the Taub-NUT version of the BBMB hairy
black hole, with or without a cosmological constant. We show that the anti-de
Sitter hyperbolic version of this solution is free of closed timelike curves
that plague usual Taub-NUT metrics, and thus consists of a rotating,
asymptotically locally anti-de Sitter black hole. This stationary solution has
no curvature singularities whatsoever in the conformal frame, and the NUT
charge is shown here to regularize the central curvature singularity of the
corresponding static black hole. Given our findings we discuss the anti-de
Sitter hyperbolic version of Taub-NUT in four dimensions, and show that the
curvature singularity of the NUT-less solution is now replaced by a neighboring
chronological singularity screened by horizons. We argue that the properties of
this rotating black hole are very similar to those of the rotating BTZ black
hole in three dimensions.
