and always assume $(H1)$ only. Let $\rho=(\rho_1,...,\rho_n)$ and $$\label{12jun20e1}
 \Lambda_J(\rho)=8\pi \sum_{i\in J} \rho_i-\sum_{i,j\in J}a_{ij}\rho_i\rho_j$$ for any $J\subset I:=\{1,...,n\}$. Define $$\Gamma_1=\{\rho; \,\, \Lambda_I(\rho)=0\quad \mbox{ and }\Lambda_J(\rho)>0 \mbox{ for all } \emptyset \subsetneq J \subsetneq I \,\, \}.$$ Note that if $(H2)$ also holds also, then $\Lambda_I(\rho)=0$ implies $\Lambda_J(\rho)>0$ for all $J\subsetneq I$ ( see [@linzhang2]). For any $\rho$ we define $(m_1,...,m_n)$ by $$\label{12apr23e1}
 m_i=\frac 1{2\pi} \sum_{j\in I} a_{ij}\rho_j.$$ The quantity $m_i$ can be interpreted by the entire solution $\tilde U$ of (\[12apr23e2\]). In fact $$\rho_i=\int_{\mathbb R^2} e^{\tilde U_i}dx, \quad i\in I,$$ and $$\label{12apr23e3}
 \tilde U_i(x)=-m_i \log |x| +O(1),\quad \mbox{ for $|x|$ near infinity. }$$ The integrability of $e^{\tilde U_i}$ implies $m_i>2$ for all $i$. On the other hand $\Lambda_I(\rho)=0$ can be written as $\sum_{i\in I}(m_i-4)\rho_i=0$. Thus either $\min\{m_1,...,m_n\}<4$ or $m_i=4$ for all $i\in I$. We also note that (\[12apr23e3\]) implies that $\Gamma_1$ is a smooth submanifold because the normal vector at $\rho\in \Gamma_1$: $$(\sum_{j\in I} a_{ij} \rho_j-4\pi,...,\sum_{j\in I} a_{nj}\rho_j -4\pi),$$ has all its components positive.

The asymptotic behavior of $\tilde U_i(x)$ shows that the decay rate of $e^{\tilde U_i(x)}$ is $O(|x|^{-m})$, where $$m=\min\{m_1,...,m_n\}.$$ In this article we define $Q\in \Gamma_1$ with $m=4$, i.e. $m_i=4$ for all $i$. Thus the decay rate 