Lensing effect of a cosmic string in Chern-Simons gravity

It is pointed out that any conformally transformed of a flat space-time metric $\ball g_{ij} = f(x)\;\eta_{ij}$ is a solution to Witten's equation of Chern-Simons gravity, which holds outside matter in (2+1) dimensions. It is also shown that a simultaneous exterior solution to both Witten's and Einstein's equations, yields the lensing effect of an isolated cosmic string, if $f(x)$ is reduced to an arbitrary dimensionless constant. However, the solution to Witten's equation with $f(x)$ being an arbitrary and continuous function of space-time coordinates, also leads to an open circular path for a light ray near the string.