# Loops of Infinite Order and Toric Foliations

*2022, Volume 27, Number 3, pp. 320-332*

Author(s):

**Efstathiou K., Lin B., Waalkens H.**

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.

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