Loops of Infinite Order and Toric Foliations

In 2005 Dullin et al. proved that the non-zero vector of Maslov indices is an eigenvector with eigenvalue 1 monodromy matrices integrable Hamiltonian system. We take a close look at geometry behind this result and extend it to more general context. construct bundle morphism defined on lattice (general) system, which can be seen as generalization indices. The non-triviality implies existence common eigenvectors matrices, gives rise corank toric foliation refining original one induced by Furthermore, we show in case where system has 2 degrees freedom, global free S^{1} action.