  Using the result of a matrix model computation of the exact glueball
superpotential, we investigate the relevant mass perturbations of the
Leigh-Strassler marginal ``q'' deformation of N=4 supersymmetric gauge theory.
We recall a conjecture for the elliptic superpotential that describes the
theory compactified on a circle and identify this superpotential as one of the
Hamiltonians of the elliptic Ruijsenaars-Schneider integrable system. In the
limit that the Leigh-Strassler deformation is turned off, the integrable system
reduces to the elliptic Calogero-Moser system which describes the N=1^* theory.
Based on these results, we identify the Coulomb branch of the partially
mass-deformed Leigh-Strassler theory as the spectral curve of the
Ruijsenaars-Schneider system. We also show how the Leigh-Strassler deformation
may be obtained by suitably modifying Witten's M theory brane construction of
N=2 theories.
