I am reading the book Lectures on the geometry of Poisson manifolds, by Izu Vaisman.
To a Poisson structure $\{\cdot,\cdot\}$ on a manifold $M$ we associate the Poisson bivector field $w\in\Gamma(\Lambda^2TM)$. We have
Lemma: $[w,w]=0$
This follows directly from proposition 1.4 in the book. Here $[\cdot,\cdot]$ is the Schouten-Nijenhuis bracket.
Now let $w$ be given locally by $w=w_1\wedge w_2$. We compute $$\begin{align} [w,w] = & [w,w_1\wedge w_2]\\ = & [w,w_1]\wedge w_2 - w_1\wedge[w,w_2]\\ = & (-w_2\wedge[w_1,w_1] + w_1\wedge[w_2,w_1])\wedge w_2\\ & -w_1\wedge(-w_2\wedge[w_1,w_2] + w_1\wedge[w_2,w_2])\\ = & w_1\wedge[w_1\wedge w_2]\wedge w_2 + w_1\wedge w_2\wedge[w_1,w_2]\\ = & 2w\wedge[w_1,w_2] \end{align}$$ This tells us that, whenever it is non-zero, $w$ locally spans an integrable distribution (by Frobenius' theorem, as we must have $[w_1,w_2] = 0$). In particular, if the Poisson structure comes from a symplectic structure then it is nowhere zero, and so we get a $2$-foliation of the manifold. Is this foliation studied somewhere? What is known about it?
Edit: I committed a rookie's error: in general $w$ is not of the form $w_1\wedge w_2$, but it is a sum of such pure terms.
My question becomes: are there interesting examples of Poisson structures on manifolds of the form we assumed above? Also, given a $2$-foliation of a manifold, is it possible to associate (canonically, if possible) a Poisson structure to it by considering the tangent space of the leaves?