\label{invC}\end{aligned}$$ To derive these expressions, we have used the field equations, the quantities (\[RT\_R rr\])–(\[RT\_R xx\]), (\[Bach rr\])–(\[Bach xx\]), (\[WeyliK\])–(\[WeylfK\]), and relations (\[contraEinstein-WeylBHC\]), (\[confrel\]), (\[OmBach\]) together with ${C_{abcd}\,C^{abcd}=\Omega^{-4}\, C_{abcd}^{{\hbox{\tiny Kundt}}}\, C^{abcd}_{{\hbox{\tiny Kundt}}}}$ which follows from the invariance of the Weyl tensor under conformal transformations.

It is interesting to observe from (\[invB\]) and (\[Bach rr\])–(\[Bach xx\]) with (\[OmBach\]) that B\_[ab]{}=0B\_[ab]{}B\^[ab]{} =0. \[Bach=0iffINV=0\] Moreover, C\_[abcd]{}C\^[abcd]{}=0B\_[ab]{} =0, \[Weylinv=0thenBach=0\] because the relation ${{\cal H}'' +2=0}$ substituted into gives ${B_{ab}\,B^{ab} =0}$, i.e., ${B_{ab} =0}$ due to .

Notice also that the *first Bach component* ${\B_1=\H \H''''}$ *always vanishes on the horizon* where ${\H=0}$, see the condition .

In view of the key invariant , there are *two geometrically distinct classes of solutions* to (\[Eq1\]), (\[Eq2\]), depending on the Bach tensor ${B_{ab}}$. The first simple case corresponds to ${B_{ab}=0}$, while the much more involved second case, not allowed in General Relativity, arises when ${B_{ab}\ne0}$. This invariant classification has geometrical and physical consequences. In particular, the distinction of spacetimes with ${B_{ab}=0}$ and with ${B_{ab}\ne0}$ can be detected by measuring geodesic deviation of test particles, see Section \[geodeviation\] below.

${B_{ab}=0}$: Uniqueness of Schwarzschild {#integration:Schw}
-----------------------------------------

First, let us assume the metrics (\[BHmetric\]) such that ${B_{ab}=0}$ everywhere. In view 