The Painleve analysis and construction of solutions for the generalized Henon-Heiles system

摘要

The generalized Henon-Heiles system has been considered. In two nonintegrable cases new two-parameter solutions have been obtained in terms of elliptic functions. These solutions generalize the known one-parameter solutions. In these nonintegrable cases with the help of the Painleve test three-parameter special solutions have been found as Laurent series, converging in some ring. One of parameters determines the singularity point location, other parameters determine coefficients of series. For some values of these parameters the Laurent series solutions coincide with the Laurent series of the elliptic solutions, obtained in this paper. The Painleve test can be used not only to construct local solutions as the Laurent series, but also to find elliptic solutions.