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Glorfindel
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It isn’t true, take any integration of a non-twisted Dirac structure which is not Poisson and that is quasi-symplectic but non symplectic with Omega=0$\Omega=0$.

It isn’t true, take any integration of a non-twisted Dirac structure which is not Poisson and that is quasi-symplectic but non symplectic with Omega=0.

It isn’t true, take any integration of a non-twisted Dirac structure which is not Poisson and that is quasi-symplectic but non symplectic with $\Omega=0$.

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It isn’t true, take any integration of a non-twisted Dirac structure which is not Poisson and that is quasi-symplectic but non symplectic with Omega=0.