On families of lagrangian tori on hyperkähler manifolds

This is a note on Beauville’s problem (solved by Greb, Lehn, and Rollenske in the non-algebraic case, and by Hwang and Weiss in general) whether a lagrangian torus on an irreducible holomorphic symplectic manifold is a fiber of a lagrangian fibration. We provide a different very short solution in the non-algebraic case, and make some observations suggesting a different approach in the algebraic case.