  We consider non-topological solutions of a nonlinear elliptic system problem
derived from the $SU(3)$ Chern-Simons models in $\mathbb{R}^2$. The existence
of non-topological solutions even for radial symmetric case has been a long
standing open problem. Recently, [Choe, Kim, Lin (2015, 2016)] showed the
existence of radial symmetric non-topological solution when the vortex points
collapse. However, the arguments in [Choe, Kim, Lin (2015, 2016)] cannot work
for an arbitrary configuration of vortex points. In this paper, we develop a
new approach by using different scalings for different components of the system
to construct a family of non-topological solutions, which blows up at infinity.
