\alpha/\gamma= G\alpha}$. For ${k=0}$ we recover vacuum Einstein’s equations of General Relativity. Interestingly, all solutions of (\[fieldeqsEWmod\]) in *Einstein–Weyl gravity* (${\beta=0}$) with ${R=0}$ *are also solutions to general Quadratic Gravity* (${\beta\ne0}$) since for ${\Lambda=0}$ the QG parameter $\beta$ does not contribute to the constant $k$ defined by (\[fieldeqsgen\]).

Black hole metrics {#BHmetricsec}
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For studying static, nonrotating black holes, it is a common approach to employ the canonical form of a general spherically symmetric metric $${{\rm{d}}}s^2 = -h(\bar r)\,{{\rm{d}}}t^2+\frac{{{\rm{d}}}\bar r^2}{f(\bar r)}+\bar r^2({{\rm{d}}}\theta^2+\sin^2\theta\,{{\rm{d}}}\phi^2) \,.
\label{Einstein-WeylBH}$$ In particular, for the famous *Schwarzschild solution* of Einstein’s General Relativity [@Schwarzschild:1916] (and also of QG), the two metric functions *are the same* and take the well-known form $$f(\bar{r}) = h(\bar{r})=1-\frac{2m}{\bar{r}} \,.
\label{SchwarzschildBH}$$ The metric (\[Einstein-WeylBH\]) was also used in the seminal papers [@LuPerkinsPopeStelle:2015; @LuPerkinsPopeStelle:2015b] to investigate generic spherical black holes in Quadratic Gravity, in which it was surprisingly shown, mostly by numerical methods, that such a class contains further black-hole solutions *distinct* from the Schwarzschild solution (\[SchwarzschildBH\]). It turned out that while the Schwarzschild black hole has ${f=h}$, this non-Schwarzschild black hole is characterized by ${f\not=h}$. However, due to the complexity of the QG field equations (\[GenQGFieldEq\]) for the classical metric form (\[Einstein-WeylBH\]), it has 