Some exact solutions of the semilocal Popov equations

We study the semilocal version of Popov's vortex equations on $S^2$. Though they are not integrable, we construct two families of exact solutions which are expressed in terms of rational functions on $S^2$. One family is a trivial embedding of Liouville-type solutions of the Popov equations obtained by Manton, where the vortex number is an even integer. The other family of solutions are constructed through a field redefinition which relate the semilocal Popov equation to the original Popov equation but with the ratio of radii $\sqrt{3/2}$, which is not integrable. These solutions have vortex number $N=3n-2$ where $n$ is a positive integer, and hence $N=1$ solutions belong to this family. In particular, we show that the $N=1$ solution with reflection symmetry is the well-known $CP^1$ lump configuration with unit size where the scalars lie on $S^3$ with radius $\sqrt{3/2}$. It generates the uniform magnetic field of a Dirac monopole with unit magnetic charge on $S^2$.