parameter $m$ is the Schwarzschild mass*.

Each strictly monotonic[^1] function $\phi(r)$ of class $C^2\big([0,\infty)\big)$ with the asymptotic behaviour $$\label{phi}
\phi= O\!\left(r^{-1/2-\textstyle{\alpha}}\right)\!,\;\;\, r\rightarrow\infty\qquad (\alpha>0)$$ determines a one-parameter family of solutions by these quadratures: one sequentially finds the functions $\mathrm{e}^{\Phi}$, $\xi$, $A$, $f$, and $\widetilde{V}(r)$, and then find the potential $V(\phi)=\widetilde{V}(r(\phi))$. Note also that we could include the cosmological constant in the potential as the additional term $\Lambda/2$, but its contribution to the geometry of the central region would be negligible. The absence of the cosmological constant simply means that $V(\phi(\infty))=0$.

In spherically symmetric spacetimes, the Kretchmann invariant, $K= R_{\alpha\beta\gamma\delta} R^{\alpha\beta\gamma\delta}/4$, equals the sum of the squared curvature components. In the orthonormal basis associated with the metric (\[metric\]), algebraically independent components of the curvature can be reduced with the aid of (\[F-xi\]) and (\[A-f\]) to the form $$\begin{aligned}
R_{0101}&=& {\phi'}^{2}f-\frac{f-1}{r^2}, \quad R_{2323}=\frac{f-1}{r^2},
\label{R1}\\
R_{0202}&=&R_{0303}= -\frac{f}{r^2}+ \mathrm{e}^{-\Phi}\,\frac{\xi-3m}{r^3},
\label{R2}\\
R_{1212}&=&R_{1313}=\frac{f}{r^2}- {\phi'}^{2}f- \mathrm{e}^{-\Phi}\,\frac{\xi-3m}{r^3},
\label{R3}\end{aligned}$$ One can see that with the exception of some special fine-tuned cases, $K$ and $R$ diverge at $r=0$ for all solutions. In the generally accepted manner, we call a solution *a naked singularity* (*a black hole*) if $K$ diverges at $r=0$ and $f>0$ for all $r>0$ (respectively, $f=0$ at some radius 