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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
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.