  The pure braid group \Gamma of a quadruply-punctured Riemann sphere acts on
the SL(2,C)-moduli M of the representation variety of such sphere. The points
in M are classified into \Gamma-orbits. We show that, in this case, the
monodromy groups of many explicit solutions to the Riemann-Hilbert problem are
subgroups of SU(2). Most of these solutions are examples of representations
that have dense images in SU(2), but with finite \Gamma-orbits in M. These
examples relate to explicit immersions of constant mean curvature surfaces.
