scalar for (\[Kundt seed xy\]) is ${\Psi_2=-\frac{1}{12}({\cal H}''+2)}$, and both ${\mbox{\boldmath$k$}}$ and ${\mbox{\boldmath$l$}}$ are double principal null directions. For the specific function (\[Schw\]), ${\Psi_2=m\,r}$. The Kundt seed geometry for the Schwarzschild solution is thus of algebraic type D. It is conformally flat if, and only if, ${m=0}$, in which case it is the Bertotti–Robinson spacetime.

The Robinson–Trautman form of the black hole metrics {#RT}
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Recently, we have proven in [@PravdaPravdovaPodolskySvarc:2017] that *any metric conformal to a Kundt geometry must belong to the class of expanding Robinson–Trautman geometries* (or it remains in the Kundt class). Indeed, performing a simple transformation ${r(\tilde r)}$ of (\[confrelation\]), (\[Kundt seed xy\]), such that $$r = \int\!\!\frac{{{\rm{d}}}\tilde r}{\Omega^2(\tilde r)}\, , \qquad
{{H}}\equiv \Omega^{2}\, \H \,,
\label{guu_RT}$$ we obtain $${{\rm{d}}}s^2_{\hbox{\tiny RT}} = \Omega^2(\tilde r)\,\frac{{{\rm{d}}}x^2 + {{\rm{d}}}y^2}{\big(1+\ctvrt(x^2+y^2)\big)^2}
-2\,{{\rm{d}}}u\,{{\rm{d}}}\tilde r+{{H}}(\tilde r)\,{{\rm{d}}}u^2 \,. \label{confRT}$$ This has the canonical form of the Robinson–Trautman class [@Stephanietal:2003; @GriffithsPodolsky:2009] with the identification \_[,r]{} = ,[[H]{}]{}= - h. The Schwarzschild black hole is recovered for ${\Omega(\tilde r)=\tilde r}$ that is ${\Omega_{,\tilde r}=1}$, equivalent to ${f(\bar{r}) = h(\bar{r})}$. Other distinct non-Schwarzschild black hole solutions are identified by ${f(\bar{r}) \ne h(\bar{r})}$. The Killing horizon is obviously given by ${{{H}}(\tilde r_h)=0}$, corresponding to ${\H(r_h)=0=h(\bar r_h)}$ and ${f(\bar 