  For any rank 2 of simple Lie algebra, the relativistic Chern-Simons system
has the following form: \begin{equation}\label{e001} \left\{\begin{array}{c}
\Delta u_1+(\sum_{i=1}^2K_{1i}e^{u_i}
-\sum_{i=1}^2\sum_{j=1}^2e^{u_i}K_{1i}e^{u_j}K_{ij})=4\pi\displaystyle
\sum_{j=1}^{N_1}\delta_{p_j}\\ \Delta u_2+
(\sum_{i=1}^2K_{2i}e^{u_i}-\sum_{i=1}^2\sum_{j=1}^2e^{u_i}K_{2i}e^{u_j}K_{ij})=4\pi\displaystyle
\sum_{j=1}^{N_2}\delta_{q_j} \end{array} \right.\mbox{in}\; \mathbb{R}^2,
\end{equation} where $K$ is the Cartan matrix of rank $2$. There are three
Cartan matrix of rank 2: ${\bf A}_2$, ${\bf B}_2$ and ${\bf G}_2$. A
long-standing open problem for \eqref{e001} is the question of the existence of
non-topological solutions. In a previous paper \cite{ALW}, we have proven the
existence of non-topological solutions for the ${\bf A}_2$ and ${\bf B}_2$
Chern-Simons system. In this paper, we continue to consider the ${\bf G}_2$
case. We prove the existence of non-topological solutions under the condition
that either $N_2\displaystyle\sum_{j=1}^{N_1} p_j=N_1\displaystyle
\sum_{j=1}^{N_2} q_j $ or $N_2\displaystyle\sum_{j=1}^{N_1}p_j \not
=N_1\displaystyle \sum_{j=1}^{N_2} q_j$ and $N_1,N_2>1$, $
  |N_1-N_2|\neq 1$. We solve this problem by a perturbation from the
corresponding ${\bf G}_2$ Toda system with one singular source. Combining with
\cite{ALW}, we have proved the existence of non-topological solutions to the
Chern-Simons system with Cartan matrix of rank $2$.
