Reconstruction through a coisotropic reduction: the missing momentum on a Poincaré section as a branched covering

Determining the momentum missing on a Poincaré section of a two-degree-of-freedom Hamiltonian is usually posed as the root-finding problem $H(0,q_2,p_1,p_2)=E$, which degenerates at the boundary of the energetically allowed region. We argue that the degeneration is the visible symptom of a geometric