
SIGMA 2 (2006), 064, 4 pages nlin.SI/0408027
http://dx.doi.org/10.3842/SIGMA.2006.064
On a 'Mysterious' Case of a Quadratic Hamiltonian
Sergei Sakovich
Institute of Physics, National Academy of Sciences, 220072 Minsk, Belarus
Received June 02, 2006, in final form July 18, 2006; Published online July 28, 2006
Abstract
We show that one of the five cases of a quadratic
Hamiltonian, which were recently selected by Sokolov and Wolf who
used the KovalevskayaLyapunov test, fails to pass the
Painlevé test for integrability.
Key words:
Hamiltonian system; nonintegrability; singularity analysis.
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References
 Sokolov V.V., Wolf T., Integrable quadratic classical Hamiltonians
on so(4) and so(3,1), J. Phys. A: Math. Gen., 2006,
V.39, 19151926, nlin.SI/0405066.
 Ablowitz M.J., Ramani A., Segur H., A connection between
nonlinear evolution equations and ordinary dif ferential
equations of Ptype. I, J. Math. Phys., 1980, V.21,
715721.
 Ramani A., Grammaticos B., Bountis T., The Painlevé
property and singularity analysis of integrable and nonintegrable
systems, Phys. Rep., 1989, V.180, 159245.
 Tsiganov A.V., Goremykin O.V., Integrable systems on so(4) related with XXX
spin chains with boundaries, J. Phys. A: Math. Gen., 2004,
V.37, 48434849, nlin.SI/0310049.
 Sokolov V.V., On a class of quadratic Hamiltonians on so(4), Dokl. Akad. Nauk,
2004, V.394, 602605 (in Russian).
 Ramani A., Dorizzi B., Grammaticos B., Painlevé conjecture revisited,
Phys. Rev. Lett., 1982, V.49, 15391541.
 Grammaticos B., Dorizzi B., Ramani A., Integrability of
Hamiltonians with third and fourthdegree polynomial potentials,
J. Math. Phys., 1983, V.24, 22892295.
 Ablowitz M.J., Clarkson P.A., Solitons, nonlinear evolution equations and inverse
scattering, Cambridge, Cambridge University Press, 1991.
 Sakovich S.Yu., Tsuchida T., Symmetrically coupled higherorder
nonlinear Schrödinger equations: singularity analysis and
integrability, J. Phys. A: Math. Gen.,
2000, V.33, 72177226, nlin.SI/0006004.
 Sakovich S.Yu., Tsuchida T., Coupled higherorder
nonlinear Schrödinger equations: a new integrable case via the
singularity analysis, nlin.SI/0002023.

