Another example of hairy BH’s with the property that the masses $M_{\mathrm{hair}}$ scale also linearly with $r_{\Delta }$ are the hairy BH’s of the Einstein-Skyrme system. To see this we take the equations of motion for the Einstein-Skyrme model as described in the subsection (7.4) from Ref. [11]; specifically, we take the equation (7.37): $$\frac{dm(r)}{dr}=N[\frac{r^2}{2}+{sin}^2\chi ](\frac{d\chi }{dr}{)}^2+[r^2+\frac{{sin}^2\chi }{2}]\frac{{sin}^2\chi }{r^2}$$(46)where the metric for static and spherically symmetric configurations is: $$d{s}^2=-{\sigma }^2(r)N(r)d{t}^2+\frac{1}{N(r)}d{r}^2+r^{2}d{\Omega }^2$$(47)with $$N(r)=1-2m(r)/r,$$(48)and the Skyrmion field $\chi (r)$ depending on the dimensionless coordinate $r$; to continuation we define a new coordinate as $x=r/r_{\Delta }$ and after integration rewrite (46) as $$\begin{gathered} M_{\mathrm{ADM}}(r_{\Delta })=\frac{r_{\Delta }}{2}+M_{\mathrm{hair}}(r_{\Delta }) \\ =\frac{r_{\Delta }}{2}+r_{\Delta }{\int }_1^{\infty }\left[\frac{N{x}^2}{2}\right(\frac{d\chi }{dx}{)}^2+{sin}^2\chi ]dx \\ +\frac{1}{r_{\Delta }}{\int }_1^{\infty }[N(\frac{d\chi }{dx}{)}^2+\frac{{sin}^2\chi }{2x^2}]({sin}^2\chi )dx \end{gathered}$$(49)The behavior for $\chi (x)$ can see from left panel in the Fig. 14. of such reference. From equation (7.41) we see the asymptotic behavior for $\chi (x)$ as $x\to \infty$ (with $a$ a constant): $\chi \approx a{x}^{-2}$. Although this analysis shows that $M_{\mathrm{ADM}}(r_{\Delta })$ grows linearly with $r_{\Delta }$ for $r_{\Delta }\gg 1$, we can not apply the limit $r_{\Delta }\to \infty$ to (49) because the existence of Skyrme’s black holes is limited to configurations with horizon radius $r_{\Delta }\lesssim r_{\Delta }^{\max }(\kappa )$ (here $\kappa \equiv 4\pi G_0{f}^2$ is the coupling constant of the theory and $r_{\Delta }^{\max }(\kappa )$ is a maximal value depending of $\kappa$) and $\kappa \lesssim {\kappa }_{bh}^{\max }$. The value of ${\kappa }_{bh}^{\max }$ is of the order of 0.0315. At the other hand, the existence of solitons is permitted for $\kappa \lesssim 0.0437$. In the Einstein-Higgs system at hand, $r_{\Delta }$ seems to be limited by the precision of the shooting method. For large $r_{\Delta }$, the shooting parameter ${\varphi }_{\Delta }$ approaches a below limit ($\approx 0.26$ for the model ${\overrightarrow{P}}_{0.1}$ of Fig. 10). It is unknown if this limit for the SHBH configurations is a fundamental limit or not. It turns then interesting to construct a similar model based in the isolated-horizon formalism for the Einstein-Skyrme system to compare similarities and differences with the model presented here [17].