Flatness of generic Poisson pairs in odd dimension

Given a $(m-2)$-form $\text{zw}$ and a volume form $\text{zW}$ on a $m$-manifold one defines a bi-vector $\text{zL}$ by setting $\zL(\za,\zb)={\frac {\za\zex\zb\zex\zw} {\zW}}$forany$1$-forms$\za,\zb$. In this way, locally, a Poisson pair, or bi-Hamiltonian structure, $(\text{zL},{\text{zL}}_1)$ is always represented by a couple of $(m-2)$-forms $\text{zw},{\text{zw}}_1$ and a volume form $\text{zW}$. Here one shows that, for $m\text{zmai}5$ and odd and $(\text{zL},{\text{zL}}_1)$ generic, $(\text{zL},{\text{zL}}_1)$ is flat if and only if there exists a $1$-form $\text{zl}$ such that $d\text{zw}=\text{zl}\text{zex}\text{zw}$ and $d\zw_1 =\zl\zex\zw_1$.