Sections of Lagrangian fibrations on holomorphic symplectic manifolds

Let M be a holomorphically symplectic manifold, equipped with a Lagrangian fibration: \; M X. A degenerate twistor deformation (sometimes also called "a Tate-Shafarevich twist") is a family of holomorphically symplectic structures on M parametrized by H^1, 1 (X). All members of this family are equipped with a holomorphic Lagrangian projection to X, and their fibers are isomorphic to the fibers of. Assume that M is a compact hyperkahler manifold of maximal holonomy, and the Lagrangian projection has no multiple fibers in codimension 1. We prove that M has a degenerate twistor deformation M' such that the Lagrangian projection: \; M' X admits a meromorphic section.