B_{ab}\ne0}$, which is in contradiction with vacuum Einstein’s equations ${R_{ab}=0}$.

In the rest of this paper, we now concentrate on these new spherical spacetimes in QG, in particular on black holes generalizing the Schwarzschild solution. First, we integrate the field equations (\[Eq1\]), (\[Eq2\]) for the metric functions $\Omega(r)$ and ${\cal H}(r)$. Actually, we demonstrate that there are several classes of such solutions with ${B_{ab}\ne0}$. After their explicit identification and description, we will analyze their geometrical and physical properties.

Solving the field equations {#integration}
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For nontrivial Bach tensor (${\B_1, \B_2 \ne0}$), the right-hand sides of the field equations (\[Eq1\]), (\[Eq2\]) are nonzero so that the nonlinear system of two ordinary differential equations for $\Omega(r)$, ${\cal H}(r)$ is coupled in a complicated way. Finding explicitly its general solution seems to be hopeless. However, *it is possible to write the admitted solutions analytically, in terms of (infinite) mathematical series expressed in powers of the radial coordinate $r$*.

In fact, there are *two natural possibilities*. The first is the expansion in powers of the parameter ${\Delta \equiv r-r_0}$ which expresses the solution around any finite value $r_0$ (including ${r_0=0}$). The second possibility is the expansion in powers of $r^{-1}$ which is applicable for 