We study the generalized Hamiltonian dynamics of an implicit Hamiltonian system considered as a Lagrangian variety in the symplectic tangent bundle. Singularities of such systems where already considered by J. Basto-Goncalves and A. Davydov. We investigate the global properties of compact, smoothly integrable Lagrangian immersions with fold singularities.
We show that the number of intersection points of an immersion with the zero section of the bundle is estimated by a doubled sum of the self-intersection numbers. Examples of the sphere and the compact orientable surface of genus 2 will be explicitly presented.