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Thank you for the answer; that makes a lot of sense. Do you have any ideas on the proof of symplecticity? (Perhaps is it a consequence of some generic result on symplectic reduction?)
Section 5.5 there was helpful. I think I had misunderstood the definition of non-degeneracy; since it is defined so that 1-forms and derivations are 1-to-1, central elements are indeed locally constant.
