We consider symplectic manifolds with Hamiltonian torus actions which are âalmost 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 âcentered" and the moment map is proper. In particular, this classifies the preimages under the moment map of all sufficiently small open sets, which is an important step towards global classification. As an application, we construct a full packing of each of the Grassmannians Gr⁺(2, R⁵) and Gr⁺(2, R⁶) by two equal symplectic balls.
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Karshon et al. (2001) studied this question.