  We consider the Chern-Simons gauge theory of rank $2$ such as $SU(3)$,
$SO(5)$, and $G_2$ Chern-Simons model in $\mathbb{R}^2$. There may exist three
types of solutions in these theories, that is, topological, nontopological, and
mixed type solutions. Among others, mixed type solutions can only exist in
non-Abelian Chern-Simons models. We show the existence of mixed type solutions
with an arbitrary configuration of vortex points which has been a long-standing
open problem. To show it, as the first step, we need to find when a priori
bound would fail. For the purpose, we shall find partially blowing up mixed
type solutions by using different scalings for different components. Due to the
different scalings, we should control the mass contribution from infinity which
is one of the important parts in this paper.
