Vanishing cycles in complex symplectic geometry

We study the vanishing cycles on the Milnor fibre for some non-isolated singularities which appear naturally in symplectic geometry. Under assumptions given in the text, we show that the vanishing cycles associated to a distinguished basis freely generate the corresponding homology groups of the Milnor fibre. We derive some consequences of this fact, in particular for the study of integrable systems and of adjoint orbits in Lie algebras.