Centered complexity one Hamiltonian torus actions

We consider symplectic manifolds with Hamiltonian torus actions which are " almost but not quite completely integrable": the dimension of the torus is one less than half the dimension of the manifold. We provide a complete set of invariants for such spaces when they are "centered" and the moment map is proper. In particular, this classifies the preimages under the moment ma p of all sufficiently small open sets, which is an important step towards g lobal classification. As an application, we construct a full packing of eac h of the Grassmannians Gr(+) (2; R-5) and Gr(+) (2; R-6) by two equal sympl ectic balls.