Topology of domains of possible motions of integrable systems
Sbornik: Mathematics, 1996, vol. 187, no. 5, pp. 679–684
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A study is made of analytic invertible systems with two degrees of freedom on a fixed three-dimensional manifold of level of the energy integral. It is assumed that the manifold in question is compact and has no singular points (equilibria of the initial system). The natural projection of the energy manifold onto the two-dimensional configuration space is called the domain of possible motion. In the orientable case it is sphere with $k$ holes and $p$ attached handles. It is well known that for $k=0$ and $p\geqslant 2$, the system possesses no non-constant analytic integrals on the corresponding level of the energy integral. The situation in the case of domains of possible motions with a boundary turns out to be very different. The main result can be stated as follows: there are examples of analytically integrable systems with arbitrary values of $p$ and $k\geqslant 1$.
Kozlov V. V., Ten V. V., Topology of domains of possible motions of integrable systems, Sbornik: Mathematics, 1996, vol. 187, no. 5, pp. 679–684