Anisov, S. S. On the topology of an integrable Hamiltonian system in the neighborhood of a degenerate one-dimensional trajectory. (Russian) Zbl 0855.58024 Tr. Semin. Vektorn. Tenzorn. Anal. 25, Pt. 1, 17-24 (1993). Let \(s\text{grad } h\) be an integrable Hamiltonian system on a compact 4-dimensional manifold \(M^4\) with a symplectic structure. Let \(f\) be its first integral, a.e. independent of \(h\), \(Q^3= \{h(x)= E\}\) an isoenergetic surface, \(F(x)=(h(x), f(x))\) a momentum map, and let \(K\) be the set of singular points of \(F\) and \(\Sigma= F(K)\) the bifurcation diagram. Let \(A\) be a point of a singular one-dimensional orbit \(O\) and \(U\) some neighbourhood of \(A\) in the two-dimensional manifold transversal to \(O\) in \(Q\). Denote by \(\Phi\) the Poincaré return map on \(U\). Then using the classification of \(d_A\Phi\) \((\text{det } d_A \Phi=1)\) and some invariance one obtains (under some mild assumption) that \(d_A \Phi^q=\text{id}\) with \(q\) less or equal to the degree of \(f\) at \(A\). Further, in some cases there exists a unique power series \(H\) so that \(\Phi\) is a time 1 map of the vector field \(s\text{grad } H\). Reviewer: T.Nowicki (Warszawa) MSC: 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems 54H20 Topological dynamics (MSC2010) Keywords:degenerate orbit; integrable Hamiltonian system; Poincaré return map PDFBibTeX XMLCite \textit{S. S. Anisov}, Tr. Semin. Vektorn. Tenzorn. Anal. 25, Part 1, 17--24 (1993; Zbl 0855.58024)