As above, let v = sgradH be an integrable Hamiltonian system restricted

to the compact isoenergy surface Q

, and let W

be its marked molecule.

Consider an arbitrary edge e of the molecule W

. Recall that it represents

a one-parameter family of tori. Suppose that, on some Liouville torus from this

family, we have chosen and xed an arbitrary basis in its fundamental group,

i.e., a pair of cycles (; ). According to the Liouville theorem, the trajectories

of the Hamiltonian system on the torus are windings (rational or irrational). This

means that there exists a coordinate system


mod 2; '

mod 2)

on the torus in which v is straightened and takes the form

v = a



+ b




Moreover, the coordinate lines of this coordinate system f'

= constg and


= constg are homologous to the basis cycles and , respectively.