Hyper-symplectic structures on integrable systems (Q1781421)

Let \(X\) be a symplectic manifold with fibres Lagrangian tori. On a fibration \(X\rightarrow B\), a flat torsion-free connection is defined. The main results here state that if \(B\) has a covariantly constant symplectic structure, then \(X\) admits a hyper-symplectic structure and \(B\) is a special Kähler manifold. In particular, there exists a bi-Lagrangian immersion of \(B\) into \(X\).