,$$ where $d\Sigma_{m-2}^2$ is the line element of $(m-2)$-dimensional surface with constant scalar curvature $(m-2)(m-3)\delta$. Without loss of generality, $\delta$ can be set to $\pm1$ or zero. It is easy to find that the nonvanishing components of the Riemann tensor are $$R^{tr}{}{}_{tr}=-\frac{f^{''}}{2},\quad
R^{ti}{}{}_{tj}=R^{ri}{}{}_{rj}=-\frac{f^{'}}{2r}\delta_{j}^i,\quad
R^{ij}{}{}_{kl}=\frac{\delta-f}{r^2}\delta_{kl}^{ij}\, ,$$ where the prime stands for derivative with respect to $r$. The Euler density then has the form $$\begin{aligned}
\bar{L}_i&=&
\frac{(m-2)!}{(m-2-2i)!}\left(\frac{\delta-f}{r^2}\right)^{i}+4i
\frac{(m-2)!}{(m-1-2i)!} \left(-\frac{f^{'}}{2r}\right)
\left(\frac{\delta-f}{r^2}\right)^{i-1}
\nonumber \\
&+&2i \frac{(m-2)!}{(m-2i)!}
\left(-\frac{f^{''}}{2}\right)\left(\frac{\delta-f}{r^2}\right)^{i-1}
+4i(i-1)\frac{(m-2)!}{(m-2i)!}\left(-\frac{f^{'}}{2r}\right)^2
\left(\frac{\delta-f}{r^2}\right)^{i-2}\, .\nonumber\\\end{aligned}$$ Defining $$F(r)=\frac{\delta-f(r)}{r^2}\, ,$$ $\bar{L}_i$ can be rewritten as $$\bar{L}_i = \frac{(m-2)!}{(m-2i)!}\frac{1}{r^{m-2}}\left(r^m
F(r)^i\right)^{''}\, .$$ Finally Eq. (\[2eq14\]) becomes $$\label{2eq21}
 0=\sum_{i=0}^{s}\widehat{\alpha}_{i}\left(r^m
F(r)^i\right)^{''}\, ,$$ where $$\widehat{\alpha}_{i}=\frac{(m-2)!}{(m-2i)!}\alpha_i\, .$$ The solution to (\[2eq21\]) is determined by the algebraic equation $$\label{Fequ}
\sum_{i=0}^{s}\widehat{\alpha}_{i}F(r)^i=\frac{M}{r^{m-1}}+\frac{Q}{r^{m}}\,
,$$ where $M, Q$ are two integration constants. Naively one may think they are related to the mass and Weyl charge of the solution [@Maeda:2006iw; @Dadhich:2000am; @Shiromizu:1999wj], respectively. But actually the integration constant $M$ has nothing to do with the mass of the solution, which will be shown shortly. The constant $Q$ may be positive, zero and negative. Here some remarks are in order.

\(i) Since $\mathcal{G}^{a}_{b}\sim \delta^{a}_{b}$, the equations $\mathcal{G}^{a}_{b}=0$ reduce to a single equation (\[2eq14\]). This is very different from the normal case in Lovelock gravity, 