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I’ve been thinking about the algebro-geometric meaning of the Lenard-Magri scheme of getting an integrable system from a pair of compatible Poisson structures. I think one might be able to prove a general result to this effect by considering the corresponding family of Poisson varieties over the projective line with an $\mathcal O(1)$-valued Poisson structure, and various subfamilies of symplectic leaves (carrying an $\mathcal O(-1)$-valued symplectic form). (It looks like the only assumption needed is that the generic rank of the Poisson structure is constant in the family.)

Do you know of any references where something like this is implemented and/or a general integrability result in this setting is proved?

EDIT: The condition I suggested in the original question is actually insufficient, and the simplest counterexample is $\mathbb A^2=\mathrm{Spec}\,\Bbbk[x,y]$ with $\{x,y\}_1=1,\ \{x,y\}_2=x$. So perhaps I should clarify the argument I had in mind for proof of the integrability. The main statement is as follows:

Lemma. Let $\pi\colon X\to\mathbb P^1$ be a smooth family with an $\mathcal O(-1)$-valued symplectic structure $\omega\colon {\wedge}^2\mathcal T_{X/\mathbb P^1}\to\pi^*\mathcal O_{\mathbb P^1}(-1)$. Consider $L=\mathrm{Sect}(\mathbb P^1,X)$, the scheme of sections of $\pi$, so that for any scheme $S$, we have a bijection $\mathrm{Hom}(S,L)\overset\sim\to \mathrm{Hom}_{\mathbb P^1}(\mathbb P^1\times S,X)$ functorial in $S$. (It exists by virtue of $\mathbb P^1$ being proper.) Then for any $\lambda\in\mathbb P^1$, the image of $L$ in $X_\lambda:=X\times_{\mathbb P^1}\{\lambda\}$ Is a smooth Lagrangian subvariety.

I still think this lemma is true, and it can be proved using Serre duality. Furthermore, by definition, a pair of compatible Poisson brackets on $Y$ give an $\mathcal O(1)$-valued fiberwise Poisson structure on the trivial family $Y \times\mathbb P^1 \to\mathbb P^1$, which can be restricted to subfamily of symplectic leaves through a given point, provided that these leaves all have the same dimension. However, it might happen, as in the example above, that there is no point with such property! One should therefore add this additional assumption in the original integrability criterion.

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