Hamiltonian stationary Lagrangian surfaces in C2

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.