The $(2+1)$-dimensional relativistic Chern-Simons equations form a nonlinear system of partial differential equations for a gauge field $A_\mu$ and a Higgs field $\varphi$ defined on ${\Bbb R}^3$ with standard Lorentzian metric. The self-dual solutions absolutely minimize the energy. There are two possible boundary conditions $|\varphi(x)|\to 1$ or $|\varphi(x)|\to 0$ as ${\Bbb R}^2\ni x\to\infty$ consistent with finite energy. Solutions with $|\varphi(x)|\to 1$ have been dubbed `topologicalâ€™ and were shown to exist by {\it R. Wang} [Commun. Math. Phys. 137, No. 3, 587-597 (1991;

Zbl 0733.58009)].
In this article, the authors consider the existence of self-dual `non-topologicalâ€™ solutions, i.e. with boundary condition $|\varphi(x)|\to 0$. They prove the existence of solutions with arbitrarily prescribed zeroes for the Higgs field and other good properties. In particular, these solutions are not in any way symmetric. The construction is obtained by perturbation about explicit solutions of the Liouville equation.