Lagrangians, symplectomorphisms and zeroes of moment maps

I will present two constructions of Kähler manifolds, endowed with Hamiltonian torus actions of infinite dimension. In the first example, zeroes of the moment map are related to isotropic maps from a surfaces in ℝ2n. In the second example, which is actually a hyperKähler moment map, the zeroes are related to symplectic maps of the torus T4. The corresponding modified moment map flows have short time existence. Polyhedral analogues of these constructions can be used to investigate piecewise linear symplectic geometry.