Geodesic
Complex Hamiltonian systems on $\mathbb{C^2}$ with Hamiltonian function of low Laurent degree
Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 2 (2015), pp. 3-9 Cet article a éte moissonné depuis la source Math-Net.Ru

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We study complex Hamiltonian systems on $\mathbb C\times(\mathbb C\setminus\{0\})$ with standard symplectic structure $\omega_{\mathbb C}=dz\wedge dw$ and Hamiltonian function $f=a z^2+b/w+P_n(w)$, where $P_n(w)$ is a polynomial of degree $n$, the numbers $a,b\in\mathbb C$ and $ab\ne0$. For some natural classes of these $\mathbb C$-Hamiltonian systems we study an equivalence relation in the Hamiltonian sense and determine the topology of the corresponding quotient space. We also prove that for $\mathbb C$-Hamiltonian systems with Hamiltonian function $f=a z^2+b/w+P_n(w)$, where $a b\ne0,n\ge0$, the bifurcation complex is homeomorphic to a two-dimensional plane. 
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     author = {N. N. Martynchuk},
     title = {Complex {Hamiltonian} systems on $\mathbb{C^2}$ with {Hamiltonian} function of low {Laurent} degree},
     journal = {Vestnik Moskovskogo universiteta. Matematika, mehanika},
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     url = {http://geodesic.mathdoc.fr/item/VMUMM_2015_2_a0/}
}
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N. N. Martynchuk. Complex Hamiltonian systems on $\mathbb{C^2}$ with Hamiltonian function of low Laurent degree. Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 2 (2015), pp. 3-9. http://geodesic.mathdoc.fr/item/VMUMM_2015_2_a0/

[1] Lepskii T.A., “Nepolnye integriruemye gamiltonovy sistemy s kompleksnym polinomialnym gamiltonianom maloi stepeni”, Matem. sb., 202:10 (2010), 109–136 | DOI | MR

[2] Kudryavtseva E.A., Lepskii T.A., “Topologiya lagranzhevykh sloenii integriruemykh sistem s giperellipticheskim gamiltonianom”, Matem. sb., 202:3 (2010), 69–106 | DOI

[3] Kudryavtseva E.A., Lepskii T.A., “Topologiya sloeniya i teorema Liuvillya dlya integriruemykh sistem s nepolnymi potokami”, Tr. Seminara po vektornomu i tenzornomu analizu, 28, 2011, 106–149

[4] Kudryavtseva E.A., Lepskii T.A., “Integriruemye gamiltonovy sistemy s nepolnymi potokami i mnogougolniki Nyutona”, Sovremennye problemy matematiki i mekhaniki, 6:3 (2011), 42–55

[5] Kudryavtseva E.A., “Analog teoremy Liuvillya dlya integriruemykh gamiltonovykh sistem s nepolnymi potokami”, Dokl. RAN, 445:4 (2012), 383–385 | MR | Zbl

[6] Bolsinov A.V., Fomenko A.T., Integriruemye gamiltonovy sistemy. Geometriya, topologiya, klassifikatsiya, RKhD, Izhevsk, 1999 | MR

[7] Bolsinov A.V., Fomenko A.T., Integrable geodesic flows on two-dimensional surfaces, Kluwer Academic Plenum Publishers, N.Y.–Boston–Dordrecht–L.–M., 2000 | MR | Zbl

[8] Fomenko A.T., Konyaev A.Yu., “New approach to symmetries and singularities in integrable Hamiltonian systems”, Topol. and its Appl., 159 (2012), 1964–1975 | DOI | MR | Zbl

[9] Fomenko A.T., “Topologicheskii invariant, grubo klassifitsiruyuschii integriruemye strogo nevyrozhdennye gamiltoniany na chetyrekhmernykh simplekticheskikh mnogoobraziyakh”, Funkts. analiz i ego pril., 25 (1991), 23–35

[10] Kudryavtseva E.A., Fomenko A.T., “Gruppy simmetrii pravilnykh funktsii Morsa na poverkhnostyakh”, Dokl. RAN, 446:6 (2012), 615–617 | Zbl

[11] Kudryavtseva E.A., Nikonov I.M., Fomenko A.T., “Maksimalno simmetrichnye kletochnye razbieniya poverkhnostei i ikh nakrytiya”, Matem. sb., 199:9 (2008), 3–96 | DOI | MR | Zbl