  We study Hamiltonian stationary Lagrangian surfaces in C^2, i.e. Lagrangian
surfaces in C^2 which are stationary points of the area functional under smooth
Hamiltonian variations. Using loop groups, we propose a formulation of the
equation as a completely integrable system. We construct a Weierstrass type
representation and produce all tori through either the integrable systems
machinery or more direct arguments.
