out that the only possible solution in the class ${[n,p]=[-1,2]}$ is the Schwarzschild black hole for which the Bach tensor vanishes. Explicit Schwarzschild–Bach black holes with ${B_{ab}\ne0}$ are contained in the classes ${[0,1]}$ and ${[0,0]}$. The fourth class ${[n,p]=[1,0]}$ represents singular solutions without horizon, and it is equivalent to the class ${(s,t)=(2,2)}$ identified previously in [@Stelle:1978; @LuPerkinsPopeStelle:2015b; @PerkinsPhD].

Expansion in powers of $r^{-1}$ {#expansion_INF}
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Analogously, we may study and classify all possible solutions to the QG field equations for an asymptotic expansion as ${r\rightarrow \infty}$. Instead of , with , for very large $r$ we can assume that the metric functions $\Omega(r)$, $\mathcal{H}(r)$ are expanded in *negative powers* of $r$ as $$\begin{aligned}
\Omega(r)      {\!\!\!& = &\!\!\!}r^N   \sum_{i=0}^\infty A_i \,r^{-i}\,, \label{rozvojomegINF}\\
\mathcal{H}(r) {\!\!\!& = &\!\!\!}r^P \,\sum_{i=0}^\infty C_i \,r^{-i}\,. \label{rozvojcalHINF}\end{aligned}$$

Inserting the series (\[rozvojomegINF\]), (\[rozvojcalHINF\]) into the field equations , , it can be shown that *only 2 classes of such solutions are allowed*, namely =\[-1,3\]\^,=\[-1,2\]\^, \[2classes\] see Section \[expansiont\_INF\]. In subsequent Section \[description\_INF\], it will be shown that the class ${[N,P]=[-1,3]^\infty}$ represents the Schwarzschild–Bach black holes, whereas the class ${[N,P]=[-1,2]^\infty}$ is a specific Bachian generalization of a flat space which does not correspond to a black hole.

Discussion of solutions 