takes the explicit form $${\cal H}\Omega''+{\cal H}'\Omega'+{\textstyle \frac{1}{6}} ({\cal H}''+2)\Omega = 0 \,,
 \label{trace}$$ see (\[barR\]). Indeed, it immediately follows from , : just subtract from the derivative of the second equation the first equation multiplied by $\H'$ (and divide the result by $6\Omega'$).

It is a great advantage of our conformal approach with the convenient form of the new metric (\[BHmetric\]) that the field equations (\[Eq1\]), (\[Eq2\]) are *considerably simpler* than the previously used field equations for the standard metric . Moreover, they form an *autonomous system*, which means that the differential equations *do not explicitly depend on the radial variable $r$*. This will be essential for solving such a system, finding their analytic solution in the generic form , or , in subsequent Section \[integration\].

Fundamental scalar invariants and geometric classification {#invariants}
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For a geometrical and physical interpretation of spacetimes that are solutions to the field equations (\[Eq1\]), (\[Eq2\]), it will be crucial to investigate the behaviour of scalar curvature invariants constructed from the Ricci, Bach, and Weyl tensors themselves. A direct calculation yields $$\begin{aligned}
R_{ab}\, R^{ab} &=  16k^2\, B_{ab} B^{ab} \,, \label{invR}\\
B_{ab}\, B^{ab} &=  \tfrac{1}{72}\,\Omega^{-8}\,\big[(\B_1)^2 + 2(\B_1+\B_2)^2\big] \,,\label{invB}\\
C_{abcd}\, C^{abcd} &=  \tfrac{1}{3}\,\Omega^{-4}\,\big({\cal H}'' +2\big)^2 \,. 