= Obstruction to a Lenard-Magri recursion
The next step of a <Lenard-Magri recursion> requires $-\Pi_2dH_{n-1}$ to be a <Hamiltonian vector field> for $\Pi_1$. It must be tangent to the <symplectic leaves>, and its contraction with the leaf <symplectic form> must have a global potential, with a compatible smooth extension between leaves. Even the first step may fail: on $\mathbb R^2$, take $\Pi_1=0$, $\Pi_2=\partial_x\wedge\partial_y$ and $H_0=x$. Both <Poisson structures> are compatible, but $\Pi_1dH_1=-\Pi_2dx$ cannot hold, because its left side is zero and its right side is nonzero.
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