B&=\frac{4(r^2+z)^2A}{(r^2+2z)^2}\,,\\
\phi'^2&=\frac{r^2(r^2+2z)^2}{4z(r^2+z)^3A}\,.\end{aligned}$$ Notice the (+) sign of the kinetic term in the action which we adopt in order to have a real scalar field while it is defined as complex in [@Rinaldi:2012vy]. Also, in the limit $z\rightarrow \infty$, the scalar field becomes constant and the metric becomes Schwarzschild. Therefore, the solution constitutes a smooth deformation of the Schwarzschild spacetime. We have in our notations $$\begin{aligned}
K=-X,~G_3=0,~G_4=\frac{1}{2},~G_5=\frac{z}{2}\phi\,,\end{aligned}$$ which gives $$\begin{aligned}
{\cal F}&=\frac{3r^2+2z}{2(r^2+z)}\,,\\ 
{\cal G}&={\cal H}=\frac{r^2+2z}{2(r^2+z)}\,.\end{aligned}$$ We might conclude that the solution is stable because ${\cal F}>0$, ${\cal G}>0$ and ${\cal H}>0$ but, in fact, the S-deformation is not licit, since $S(r^*=0)=+\infty$ and therefore it is an example where the formulas derived can’t be used. We study the potential $V$ directly from which it is easy to see that $V(r^*=-\infty)=0$ and $V(r^*=0)=\frac{9l(l+1)-1}{36 z}$, and we have checked graphically that the potential is always positive (see Fig.\[fig:rinaldi\]), which implies the stability of the solution.

![Potential for the black hole solution (\[eq:Rinaldi1\],\[eq:Rinaldi2\]) from the horizon where the potential is null to its asymptotic value $(9l(l+1)-1)/36 z$.[]{data-label="fig:rinaldi"}](rinaldi)

Conclusion
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In this paper, we have studied odd-parity perturbations around static and spherically symmetric background spacetime in Horndeski gravity. We have derived the conditions of no-ghost and 