Geodesic
Classification of four-dimensional integrable Hamiltonian systems and Poisson actions of $\mathbb R^2$ in extended neighbourhoods of simple singular points. III. Realization
Sbornik. Mathematics, Tome 186 (1995) no. 10, pp. 1477-1491 
L. M. Lerman; Ya. L. Umanskii. Classification of four-dimensional integrable Hamiltonian systems and Poisson actions of $\mathbb R^2$ in extended neighbourhoods of simple singular points. III. Realization. Sbornik. Mathematics, Tome 186 (1995) no. 10, pp. 1477-1491. http://geodesic.mathdoc.fr/item/SM_1995_186_10_a5/
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The classification of integrable Hamiltonian systems in extended neighbourhoods of simple singular points, initiated in [1] and [2], is completed. This paper is a direct continuation of [1] and [2] and uses all the concepts, definitions, and results presented there. As promised in [2], the aim here is to realize all admissible types of the invariants describing the iso-energetic equivalence of integrable Hamiltonian systems with two degrees of freedom in extended neighbourhoods of simple singular points. In doing that, all admissible types of Poisson actions of the group $\mathbb R^2$ on a four-dimensional symplectic manifold are also realized. Some of the examples constructed are of a purely illustrative nature, while others are more meaningful and occur in applications.

[1] Lerman L. M., Umanskii Ya. L., “Klassifikatsiya chetyrekhmernykh integriruemykh gamiltonovykh sistem i puassonovskikh deistvii $R^2$ v rasshirennykh okrestnostyakh prostykh osobykh tochek, I”, Matem. sb., 183:12 (1992), 141–176 | MR | Zbl

[2] Lerman L. M., Umanskii Ya. L., “Klassifikatsiya chetyrekhmernykh integriruemykh gamiltonovykh sistem i puassonovskikh deistvii $R^2$ v rasshirennykh okrestnostyakh prostykh osobykh tochek, II”, Matem. sb., 184:4 (1993), 105–138 | MR | Zbl

[3] Bolsinov A. V., Matveev S. V., Fomenko A. T., “Topologicheskaya klassifikatsiya integriruemykh gamiltonovykh sistem s dvumya stepenyami svobody. Spisok sistem maloi slozhnosti”, UMN, 45:2 (1990), 49–77 | MR

[4] Kharlamov M. P., Topologicheskii analiz integriruemykh zadach v dinamike tverdogo tela, Izd-vo LGU, L., 1988

[5] Neumann C., “De problemate quodam mechanico, quod ad primam integralium ultraellipticorrum classem revocatur”, J. Reine. Angev. Math., 56 (1858), 46–63

[6] Dubrovin B. A., Krichever I. M., Novikov S. P., “Integriruemye sistemy, I”, Itogi nauki i tekhniki. Sovremennye problemy matematiki. Fundamentalnye napravleniya, 4, VINITI AN SSSR, M., 1985, 179–284 | MR

[7] Devaney R., “Transversal homoclinic orbits in an integrable system”, Amer. J. Math., 100 (1979), 631–642 | DOI | MR

[8] Lerman L. M., “Esche raz o strukture integriruemykh statsionarnykh voln dlya uravneniya Landau–Lifshitsa”, Metody kachestv. teorii i teorii bifurkatsii, Mezhvuz. tematich. sb. nauch. trudov, ed. L. P. Shilnikov, Nizhegorodskii gos. un-t, Nizhnii Novgorod, 1990, 138–160 | MR | Zbl

[9] Bobenko A. I., Tochnoe integrirovanie nelineinykh uravnenii metodom obratnoi zadachi s parametrom na ellipticheskoi krivoi, Diss. ...kand. fiz.-matem. nauk, L., 1985

[10] Sokolskii A. G., “Ob ustoichivosti avtonomnoi gamiltonovoi sistemy s dvumya stepenyami svobody v sluchae ravnykh chastot”, PMM, 38:5 (1974), 791–799 | MR | Zbl

[11] Sokolskii A. G., “Ob ustoichivosti avtonomnoi gamiltonovoi sistemy s dvumya stepenyami svobody pri rezonanse pervogo poryadka”, PMM, 41:1 (1977), 24–33 | MR | Zbl

[12] Nguen Tien Zung, Arnold–Liouville with singularities, Preprint S.I.S.S.A. 153/94/M, October 1994 – revised January 1995 | MR

[13] Bolsinov A. V., “Methods of compution of the Fomenko–Zieschang invariant”, Advances in Soviet Mathematics, 6, ed. Fomenko A. T., 1991, 147–183 | MR | Zbl