Loops of Infinite Order and Toric Foliations

In 2005 Dullin et al. proved that the nonzero vector of Maslov indices is an eigenvector with eigenvalue $1$ of the monodromy matrices of an integrable Hamiltonian system. We take a close look at the geometry behind this result and extend it to the more general context of possibly non-Hamiltonian systems. We construct a bundle morphism defined on the lattice bundle of an (general) integrable system, which can be seen as a generalization of the vector of Maslov indices. The nontriviality of this bundle morphism implies the existence of common eigenvectors with eigenvalue $1$ of the monodromy matrices, and gives rise to a corank $1$ toric foliation refining 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 $S^{1}$ action on the regular part of the system.