of and , this condition requires ${\B_1=0=\B_2}$, that is ””=0,’[H]{}”’-[[H]{}”]{}\^2 +2 =0. \[Bab=0-RHS\] Therefore, all left-hand sides and right-hand sides of equations (\[Eq1\]) and (\[Eq2\]) *vanish separately*, i.e., ”=2[’]{}\^2,’[H]{}’+3’\^2[H]{}+\^2 =0. \[Bab=0-LHS\] The first equations of (\[Bab=0-RHS\]) and (\[Bab=0-LHS\]) imply that ${\cal H}$ must be *at most cubic*, and $\Omega^{-1}$ must be *at most linear* in $r$. Using a coordinate freedom of the metric (\[BHmetric\]), without loss of generality we obtain ${\Omega=-1/r}$. The remaining equations (\[Bab=0-RHS\]), (\[Bab=0-LHS\]) then admit a unique solution $$\Omega(r)=-\frac{1}{r}\,,\qquad
{\cal H}(r) = -r^2-2m\, r^3   \,.
\label{IntegrSchwAdS}$$ Not surprisingly, this is exactly the Schwarzschild solution of General Relativity, see equation (\[Schw\]). Thus we have verified that the *Schwarzschild black hole spacetime is the only possible solution with vanishing Bach tensor*. Its corresponding scalar invariants (\[invR\])–(\[invC\]) are R\_[ab]{} R\^[ab]{} = 0 = B\_[ab]{} B\^[ab]{},C\_[abcd]{} C\^[abcd]{} = 48m\^2r\^6 . \[SchwarzInvariants\] Clearly, for ${m\not=0}$ there is a curvature singularity at ${r\to\infty}$ corresponding to ${\bar{r}=\Omega(r)=0}$.[^2]

${B_{ab}\ne0}$: New types of solutions to QG {#integration:nonSchw}
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Many other spherically symmetric vacuum solutions to Quadratic Gravity and Einstein–Weyl gravity exist when the Bach tensor is nontrivial. They are *much more involved, and do not exist in General Relativity*. Indeed, the field equations (\[fieldeqsEWmod\]) imply ${R_{ab}=4k\, 