  We study smooth SU(2) solutions of the Hitchin equations on R^2, with the
determinant of the complex Higgs field being a polynomial of degree n. When
n>=3, there are moduli spaces of solutions, in the sense that the natural L^2
metric is well-defined on a subset of the parameter space. We examine
rotationally-symmetric solutions for n=1 and n=2, and then focus on the n=3
case, elucidating the moduli and describing the asymptotic geometry as well as
the geometry of two totally-geodesic surfaces.
