The existence of topological solutions for the Chern-Simons equation with
two Higgs particles has been proved by Lin, Ponce and Yang [16]. However, both the
uniqueness problem and the existence of non-topological solutions have been left open.
In this paper, we consider the case of one vortex at origin. Among others, we prove the
uniqueness of topological solutions and give a complete study of the radial solutions, in
particular, the existence of some non-topological solutions