Classification of four-dimensional integrable Hamiltonian systems and Poisson actions of \(\mathbb{R}^ 2\) in extended neighborhoods of simple singular points. II. (English. Russian original) Zbl 0819.58018

Lerman, L. M.; Umanskij, Ya. L.

Russ. Acad. Sci., Sb., Math. 78, No. 2, 479-506 (1994); translation from Mat. Sb. 184, No. 4, 105-138 (1993).

Let \(H\) be a smooth function (Hamiltonian) and \(X_ H\) denotes the corresponding Hamiltonian vector field. Let \(K\) be another smooth function which is a first integral for \(X_ H\). The pair \((X_ H, K)\) is called an integrable Hamiltonian vector field if \(H\) and \(K\) are independent on a open dense subset of \(M\). In the first part of this work [Russ. Acad. Sci., Sb., Math. 77, No. 2, 511-542 (1994); translation from Mat. Sb. 183, No. 12, 141-176 (1992; Zbl 0812.58033)] the authors have classified the simple singular points of the action generated by \(H\) and \(K\) into four categories. There they have considered elliptic, saddle- center and saddle-focus type points. In this second part the results are stated and proved for the remaining saddle type case. The interesting problem about realization of these integrable Hamiltonian vector fields will be discussed in a separate paper.

Reviewer: I.Mladenov (Sofia)

MSC:

37J35	Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10	Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
34C05	Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
37G15	Bifurcations of limit cycles and periodic orbits in dynamical systems

Keywords:

completely integrable; singular points; integrable Hamiltonian vector fields