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Single-valued and multivalued solutions for the generalized Hénon-Heiles system with an additional nonpolynomial term. (English) Zbl 1112.37044
Mladenov, Ivaïlo M.(ed.) et al., Proceedings of the 6th international conference on geometry, integrability and quantization, Sts. Constantine and Elena, Bulgaria, June 3–10, 2004. Sofia: Bulgarian Academy of Sciences (ISBN 954-84952-9-5/pbk). 297-309 (2005).

The generalized Hénon-Heiles system with additional nonpolynomial term is considered. It is described by the Hamiltonian $H=\frac{1}{2}\left({x}_{t}^{2}+{y}_{t}^{2}+{\lambda }_{1}{x}^{2}+{\lambda }_{2}{y}^{2}\right)+{x}^{2}y-\frac{C}{3}{y}^{3}+\frac{\mu }{2{x}^{2}}$ and the corresponding system of the motion equations

${x}_{tt}=-{\lambda }_{1}x-2xy+\frac{\mu }{{x}^{3}},\phantom{\rule{2.em}{0ex}}{y}_{tt}=-{\lambda }_{2}y-{x}^{2}+C{x}^{2},$

with $\mu ,C\in ℝ$. The standard method for the search of elliptic solutions is a transformation of an initial nonlinear polynomial differential equation into a nonlinear algebraic system. It is demonstrated that the use of the Laurent-series solutions allow one to simplify the resulting algebraic system. This procedure is automatized and generalized on some type of multivalued solutions. To find solutions of the initial equation as Laurent or Puiseux series, the author uses the algorithm of the Painlevé test. Let’s remind, that the Painlevé test is an algorithm, which checks some necessary conditions for a differential equation to have the Painlevé property. A system of ODEs has the Painlevé property if its general solution has no movable critical singular point (more in detail about it see: [R. Conte, The Painlevé property. One century later. CRM Series in Mathematical Physics. New York, NY: Springer (1999; Zbl 0989.00036)].

##### MSC:
 37J35 Completely integrable systems, topological structure of phase space, integration methods 34M55 Painlevé and other special equations; classification, hierarchies 34A25 Analytical theory of ODE (series, transformations, transforms, operational calculus, etc.)