Loops of Infinite Order and Toric Foliations

In 2005 Dullin et al. proved that thenonzero vector of Maslov indices is an eigenvector with eigenvalue1 of the monodromy matrices of an integrable Hamiltonian system.We take a close look at the geometry behind this result and extendit to the more general context of possibly non-Hamiltonian systems.We construct a bundle morphism definedon the lattice bundle of an (general) integrable system, which canbe seen as a generalization of the vector of Maslov indices. The nontriviality of this bundle morphism implies the existence of common eigenvectors with eigenvalue 1of the monodromy matrices, and gives rise to a corank 1 toric foliationrefining the original one induced by the integrable system. Furthermore,we show that, in the case where the system has 2 degrees of freedom,this implies the existence of a compatible free 1 action on the regular part of the system.