scalar-hair theorem
----------------------

Various no-hair theorems were derived in the literature, where the only static spherical regular black hole is proved to be Schwarzschild. For that, the authors show that the scalar field should be trivial (i.e., constant or zero). We can, for example, cite the no hair theorem for massless scalar field [@chase], minimally coupled scalar field with arbitrary positive potential [@Heusler:1992ss; @Sudarsky:1995zg], nonminimally coupled scalar field [@Mayo:1996mv; @Bekenstein:1996pn], in Brans-Dicke theory of gravity [@Faraoni:2013iea] or more recently, for galileons [@Hui:2012qt]. Considering that we have for these models Schwarzschild as a solution with a constant or vanishing scalar field $(\phi_0)$, we conclude that $${\cal F}={\cal G}={\cal H}=2G_4(\phi_0,0)\,,$$ and therefore, Schwarzschild solution is stable in these models if $G_4(\phi_0,0)>0$. Of course, for that we need to consider the S-deformation technique is licit. We will see in the next example, a model for which we can’t use this approach.

Non-minimal derivative coupling
-------------------------------

Among the possible models of Horndeski, the coupling between the derivative of the scalar field and the Einstein tensor has also been studied. In this case the action is $$\begin{aligned}
S=\int{\rm d}^4x\sqrt{-g}\Bigl[\frac{R}{2}+\frac{1}{2}(\partial\phi)^2+\frac{z}{2}\phi G_{\mu\nu}\nabla^\mu\nabla^\nu\phi\Bigr]\,,\end{aligned}$$ and the black hole solution was found in [@Rinaldi:2012vy], which is $$\begin{aligned}
\label{eq:Rinaldi1}
A&=\frac{3}{4}-\frac{2M}{r}+\frac{r^2}{12z}+\frac{\sqrt{z}}{4r}\arctan\Bigl(\frac{r}{\sqrt{z}}\Bigr)\,,\\
\label{eq:Rinaldi2}
