# Interpretation of the Schouten bracket as an integrability condition

The Schouten bracket is an extension of the Lie bracket to multivector fields. Given a multivector field $$\Lambda$$ the vanishing of the Schouten bracket $$[\Lambda,\Lambda]=0$$ is referred to as a sort of "integrability" condition. Integrability of what exactly? In the case of a distribution of a collection of ordinary (1-)vector fields, of course, this is just the Frobenius theorem. What is the interpretation of this "integrability" for multivector fields? Tagging PDEs because even if I know nothing, I assume they are involved in the answer of this question.

One well-known example is the case of bivector fiels $$\pi$$. Then $$[\pi, \pi] = 0$$ is equivalent to say that $$\{f, g\} = \pi(df, dg)$$ is a Poisson bracket, i.e. satisfies the Jacobi identity. In this sense, it is really an integrability condition as one knows that a Poisson manifold admits a distribution by Hamiltonian vector fields, i.e.vector fields of the form $$X_f = [f, \pi]$$, which turns out to be integrable into a foliation by (symplectic) leaves. The integrability even can be seen to enter a second time as the symplectic leaves allow for Darboux charts being the local, i.e. integrated, version of the infinitesimal version of a symplectic form.