Fractional monodromy in the case of arbitrary resonances

The existence of fractional monodromy is proved for the compact Lagrangian fibration on a symplectic 4-manifold that corresponds to two oscillators with arbitrary non-trivial resonant frequencies. Here one means by the monodromy corresponding to a loop in the total space of the fibration the transformation of the fundamental group of a regular fibre, which is diffeomorphic to the 2-torus. In the example under consideration the fibration is defined by a pair of functions in involution, one of which is the Hamiltonian of the system of two oscillators with frequency ratio $m_1:(-m_2)$, where $m_1$, $m_2$ are arbitrary coprime positive integers distinct from the trivial pair $m_1=m_2=1$. This is a generalization of the result on the existence of fractional monodromy in the case $m_1=1$, $m_2=2$ published before.
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