Flatness of generic Poisson pairs in odd dimension

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Given a (m-2)-form and a volume form on a m-manifold one defines a bi-vector by setting (,)=/ for any 1-forms,. In this way, locally, a Poisson pair, or bi-Hamiltonian structure, (,₁) is always represented by a couple of (m-2)-forms,₁ and a volume form. Here one shows that, for m 5 and odd and (,₁) generic, (,₁) is flat if and only if there exists a 1-form such that d= and d₁ =₁.