A simple proof of the “geometric fractional monodromy theorem”

We present a simple proof of the “Geometric fractional monodromy theorem” (Broer–Efstathiou–Lukina 2010). The fractional monodromy of a Liouville integrable Hamiltonian system over a loop $\gamma\subset \mathbb{R}^2$ is a generalization of the classic monodromy to the case when the Liouville foliation has singularities over $\gamma$. The “Geometric fractional monodromy theorem” finds, up to an integral parameter, the fractional monodromy of systems similar to the $1:(-2)$ resonance system. A handy equivalent definition of fractional monodromy is presented in terms of homology groups for our proof.