Classifying affine structures with focus-focus singularities

Abstract

We study the singular affine structures of integrable systems with focus-focus singular fibers on the image of momentum maps. The classification of singular affine structures is equivalent to the classification of simple semitoric systems up to fiber-preserving symplectomorphisms but is not equivalent for semitoric systems with multiple pinched fibers and we give counterexamples for any number of more than one pinch on each fiber.