m_{t}n_{t}}R_{k_1l_1}{}{}^{m_1n_1}\cdots
R_{k_{t}l_{t}}{}{}^{m_{t}n_{t}}\, . \label{Lbart}$$ Note that $\mathcal{G}^a_b$ is always proportional to $\delta^a_b$, which is a crucial point to our discussions below. Let us note that if the following equations are satisfied $$\label{lineq}
0=A_t \equiv \sum_{k=t}^{p} {k \choose t} c_k
\frac{(D-m)!}{(D-m-2(k-t))!} \left(\frac{\bar{k}}{r_{0}^2
}\right)^{k-t},\quad t=0, \cdots, q\, ,$$ then the equations of motion (\[emm\]) are always trivially satisfied. These are $(q+1)$-linear equations for $c_0,\cdots,c_p $. Recall $c_1=1$ and if we consider the case with $$p=q+1=\left[\frac{m-1}{2}\right]+1\, ,$$ then the equations (\[lineq\]) indicate that $c_k (k \neq 1)$ has a unique expression in terms of $\left(\bar{k}/r_{0}^2 \right)$ and dimension $D$. When $\bar{k}=-1$ (which implies that $\mathcal{N}^n$ is a negative constant curvature space. It will be denoted by $\mathcal{H}^n$ in the following discussion.) and $D\geq m+2$, we find that all $c_k$ are positive. Some examples will be given soon.

Now we turn to the equations of $\mathcal{G}_{ab}$. Due to the fact that $\mathcal{G}_{ab}$ is proportional to $\delta_{ab}$, the equations $\mathcal{G}_{ab}=0$ reduce to a single equation $$\label{2eq14}
 0=\sum_{i=0}^{s}\alpha_i \bar{L}_i\, ,$$ where the coefficients $\alpha_i$’s are given by $$\alpha_i=\sum_{k=i}^{p} {k \choose i} c_k
\frac{(D-m-1)!}{(D-m-1-2(k-i))!} \left(\frac{\bar{k}}{r_{0}^2
}\right)^{k-i}\, ,$$ and $c_k$ are determined by solutions (\[lineq\]).

Let us further assume the $m$-dimensional metric $g_{ij}$ takes the form $$\label{m-2metric}
g=-f(r)dt^2+\frac{1}{f(r)}dr^2+r^2d\Sigma_{m-2}^2\, 