  The Coulomb branch of $N=2$ supersymmetric gauge theories in four dimensions
is described in general by an integrable Hamiltonian system in the holomorphic
sense. A natural construction of such systems comes from two-dimensional gauge
theory and spectral curves. Starting from this point of view, we propose an
integrable system relevant to the $N=2$ $SU(n)$ gauge theory with a
hypermultiplet in the adjoint representation, and offer much evidence that it
is correct. The model has an $SL(2,{\bf Z})$ $S$-duality group (with the
central element $-1$ of $SL(2,{\bf Z})$ acting as charge conjugation);
$SL(2,{\bf Z})$ permutes the Higgs, confining, and oblique confining phases in
the expected fashion. We also study more exotic phases.
