where $\mathcal{G}^{a}_{b}$ is not proportional to $\delta^{a}_{b}$ even under a spherical symmetric assumption. For example, the equation $\mathcal{G}^{t}_{t}=0$ of Lovelock theory will give a first order differential equation like [@Whee; @Cai4] $$0=\sum_{i=0}c_{i}\left(r^{m-1} F(r)^i\right)^{'}\, ,$$ in the static, spherically symmetric case. In that case, there is only one integration constant, which is nothing but the mass of the solution [@Cai4]. In the present case, one has only one traceless-like equation, which is a second order differential equation. There is therefore one more integration constant $Q$ in the present case.

\(ii) If $Q=0$, Eq. (\[Fequ\]) is very similar to the corresponding one for static, spherically symmetric black hole solutions in Lovelock theory [@Whee; @Cai4]. However, there are two obvious differences: one is that here the coefficients $\widehat{\alpha}_i$ are all fixed by $\bar{k}/r_0^2$ and $D$, while in the normal case, those coefficients are free parameters [@Whee; @Cai4]. We will show this below. The other is that in Eq. (\[Fequ\]), the range of $i$ is $[0,\cdots,
s=[m/2]]$, while in the normal case, the range is $ [0,\cdots,
[(m-1)/2]]$. Therefore when $m$ is even, $i$ can take the value $[m/2]$, which will not appear in the normal case.

\(iii) Since all $\widehat{\alpha}_0$ are fixed by 