The Gelfand-Cetlin (or Gelfand-Zeitlin) integrable system (Guillemin-Sternberg, 1983) is a collection of functions on an orbit of the adjoint action of $U(n)$. The Poisson bracket of any two of these functions is zero. The Hamiltonian vector fields of all of these functions span the tangent space to the orbit. This makes the orbit analogous to a toric manifold. However these orbits are normally not toric manifolds because their cohomology is not consistent with Danilov's theorem.

In this talk I will describe another system of Hamiltonian vector fields on the $SU(2)$ character variety associated to a compact oriented 2-manifold This is joint work with J. Weitsman (1992) following W. Goldman (1986). I will describe how our functions are moment maps for Hamiltonian torus actions on an open dense set of the character variety. I will also characterize the set where these functions are not well defined. Goldman's functions are well defined everywhere and form an integrable system on the character variety but they are not moment maps for Hamiltonian torus actions.

I will explain how symplectic reduction with respect to these Hamiltonian group actions gives rise to products of character varieties in lower genus.