  We give an identification of the triple reduced product of three coadjoint
orbits in SU(3) with a space of Hitchin pairs over a genus 0 curve with three
punctures, where the residues of the Higgs field at the punctures are
constrained to lie in fixed coadjoint orbits. Using spectral curves for the
corresponding Hitchin system, we identify the moment map for a Hamiltonian
circle action on the reduced product. Finally, we make use of results of Adams,
Harnad, and Hurtubise to find Darboux coordinates and a differential equation
for the Hamiltonian.
