not been possible to find an explicit analytic form of the metric functions ${f(\bar{r}), h(\bar{r})}$.

A new convenient metric form of the black hole geometry {#BH metric}
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As demonstrated in our previous works [@PodolskySvarcPravdaPravdova:2018; @SvarcPodolskyPravdaPravdova:2018], it is much more convenient to employ an *alternative metric form* of the spacetimes represented by (\[Einstein-WeylBH\]). This is obtained by performing the transformation $$\bar{r} = \Omega(r)\,, \qquad t = u - \int\! \frac{{{\rm{d}}}r}{\H(r)} \,, \label{to static}$$ resulting in [[[d]{}]{}]{}s\^2 = \^2(r) . \[BHmetric\] The two new metric functions $\Omega(r)$ and $\H(r)$ are related to $f(\bar r)$ and $h(\bar r)$ via simple relations h = -\^2 , f = -()\^2 , \[rcehf\] where prime denotes the derivative with respect to $r$. Of course, the argument $r$ of both functions $\Omega$ and $\H$ must be expressed in terms of $\bar{r}$ using the inverse of the relation ${\bar{r} = \Omega(r)}$.

The metric admits a *gauge freedom* given by a constant rescaling and a shift of $r$, r r+, u \^[-1]{}u . \[scalingfreedom\]

More importantly, this new black hole metric is *conformal* to a much simpler Kundt-type metric, [[[d]{}]{}]{}s\^2 =\^2(r)[[[d]{}]{}]{}s\^2\_. \[confrelation\] Indeed, ${{{\rm{d}}}s^2_{\hbox{\tiny Kundt}}}$ belongs to the famous class of *Kundt geometries*, which are nonexpanding, shear-free and 