Minimal Lagrangian tori and action-angle coordinates

We investigate which orbits of an $n$-dimensional torus action on a $2n$-dimensional toric K"ahler manifold $M$ are minimal. In other words, we study minimal submanifolds appearing as the fibres of the moment map on a toric K"ahler manifold. Amongst other questions we investigate and give partial answers to the following: (1) How many such minimal Lagrangian tori exist? (2) Can their stability, as critical points of the area functional, be characterised just from the ambient geometry? (3) Given a toric symplectic manifold, for which sets of orbits $S$, is there a compatible toric K"ahler metric whose set of minimal Lagrangian orbits is $S$?