Monodromy in two degrees of freedom integrable systems (Q1802928)

Let \(f:M^ 4\to\mathbb{R}^ 2\) be an integrable Hamiltonian system. Assume three conditions: 1.) The set of critical values of \(f\) contains an isolated point \(p\). \(f\) is singular only at one point \(x_ 0\in f^{-1}(p)\). 2.) All regular fibers \(f^{-1}(r)\) for \(r\) near \(p\) are diffeomorphic to the 2-torus. 3.) Near \(x_ 0\), \(f\) has a certain normal form. Then for any fixed regular value \(r_ 0\) near \(p\) and any simple positively oriented loop around \(p\) connecting \(r_ 0\) to itself, the monodromy operator \(H_ 1(f^{-1}(r_ 0);\mathbb{Z})\to H_ 1(f^{- 1}(r_ 0);\mathbb{Z})\) is shown to be represented by the matrix \({1\;0\choose 1\;1}\) in some appropriately chosen basis for \(H_ 1(f^{-1}(r_ 0);\mathbb{Z})\).