The affine invariant of proper semitoric integrable systems

This paper initiates the study of semitoric integrable systems with two degrees of freedom and with proper momentum-energy map, but with possibly nonproper S ¹-momentum map. This class of systems includes many standard examples, such as the spherical pendulum. To each such system we associate a subset of $R^2$, invariant under a natural notion of isomorphism and encoding the integral affine structure of the singular Lagrangian fibration, in the spirit of Delzant polygons for toric systems. (paper)