form of recurrent formulas, convenient gauge choices are found, and various aspects of the solutions are discussed. Sections \[expansiont\_INF\] and \[description\_INF\] focus on the same topics as Secs. \[expansiont\_0\] and \[description\], respectively, but this time for expansions $r \rightarrow \infty$. In Sec. \[summary\] the relation of the solutions obtained in Secs. \[expansiont\_0\]–\[description\_INF\] (including their special subcases) to the solutions given in the literature is discussed, and summarized in Table \[tab:3\]. Mathematical and physical aspects (specific tidal effects and thermodynamical quantities) of the Schwarzschild–Bach solutions are discussed in Sections \[discussion-and-figures\] and \[physics\], respectively. Finally, concluding remarks are given in Sec. \[conclusions\].

Quadratic Gravity and the Einstein–Weyl theory {#QGandEWtheory}
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Quadratic Gravity (QG) is a natural generalization of Einstein’s theory that includes higher derivatives of the metric. Its action in four dimensions contains additional quadratic terms, namely square of the Ricci scalar $R$ and a contraction of the Weyl tensor $C_{abcd}$ with itself [@Weyl1919; @Bach1921]. In the absence of matter, the most general QG action generalizing the Einstein–Hilbert action reads[@PravdaPravdovaPodolskySvarc:2017][^1] S = [[[d]{}]{}]{}\^4 x ( (R-2) +R\^2 - C\_[abcd]{} C\^[abcd]{} ), \[actionQG\] where ${\gamma=1/G}$ ($G$ is the Newtonian constant), $\Lambda$ is the cosmological constant, and $\alpha$, $\beta$ are additional QG theory parameters. 