large values of $r$. Let us now investigate both these cases.

Expansion in powers of ${\Delta \equiv r-r_0}$ {#expansio_DElta}
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It is a great advantage that , is an *autonomous system*. Thus we can find the metric functions in the form of an *expansion in powers of $r$ around any fixed value* ${r_0}$, $$\begin{aligned}
\Omega(r) {\!\!\!& = &\!\!\!}\Delta^n   \sum_{i=0}^\infty a_i \,\Delta^{i}\,, \label{rozvojomeg0}\\
\H(r)     {\!\!\!& = &\!\!\!}\Delta^p \,\sum_{i=0}^\infty c_i \,\Delta^{i}\,, \label{rozvojcalH0}\end{aligned}$$ where r-r\_0, \[DElta\] and $r_0$ is *any real constant*.[^3] In particular, in some cases this allows us to find solutions close to any black hole horizon $r_h$ by choosing ${r_0=r_h}$.

It is assumed that ${i=0, 1, 2, \ldots}$ are integers, so that the metric functions are expanded in integer steps of ${\Delta=r-r_0}$. On the other hand, the *dominant real powers* $n$ and $p$ in the expansions (\[rozvojomeg0\]) and (\[rozvojcalH0\]) *need not be* positive integers. We only assume that ${a_0\not=0}$ and ${c_0\not=0}$, so that the coefficients $n$ and $p$ are uniquely defined as the leading powers.

By inserting – into the field equations , , we prove in Section \[expansiont\_0\] that *only 4 classes of solutions of this form are allowed*, namely =\[-1,2\],=\[0,1\],=\[0,0\],=\[1,0\]. \[4classes\] In subsequent Section \[description\], it will turn 