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Let $Y$ be a smooth hypersurface in a projective irreducible holomorphicsymplectic manifold $X$ of dimension $2n$. The characteristic foliation $F$ isthe kernel of the symplectic form restricted to $Y$. Assume that $X$ isequipped with a Lagrangian fibration $\pi:X\to B$ and $Y=\pi^{-1}D$, where $D$is a hypersurface in $B$. It is easy to see that the leaves of $F$ arecontained in the fibers of $\pi$. We prove that a very general leaf is Zariskidense in a fiber of $\pi$.

Characteristic foliation on vertical hypersurfaces on holomorphic symplectic manifolds with Lagrangian Fibration