By a special symplectic connection we mean a torsion free connection which is either the Levi-Civita connection of a Bochner-Kähler metric of arbitrary signature, a Bochner-bi-Lagrangian connection, a connection of Ricci type or a connection with special symplectic holonomy. A manifold or orbifold with such a connection is called special symplectic.
We show that the symplectic reduction of (an open cell of) a parabolic contact manifold by a symmetry vector field is special symplectic in a canonical way. Moreover, we show that any special symplectic manifold or orbifold is locally equivalent to one of these symplectic reductions.
As a consequence, we are able to prove a number of global properties, including a classification in the compact simply connected case.
"Special symplectic connections." J. Differential Geom. 83 (2) 229 - 271, October 2009. https://doi.org/10.4310/jdg/1261495331