ar X iv : a st ro - p h / 05 02 35 6 v 1 1 7 Fe b 20 05 From the Laurent - series Solutions of Nonintegrable Systems to the Elliptic Solutions of them

The Painlevé test is very useful to construct not only the Laurent-series solutions but also the elliptic and trigonometric ones. Such single-valued functions are solutions of some polynomial first order differential equations. The standard methods for the search of the elliptic solutions consist of two independent steps: transformation of a nonlinear polynomial differential equation into a nonlinear algebraic system and finding of solutions of the obtained system. It has been demonstrated by the example of the generalized Hénon–Heiles system that the use of the Laurent-series solutions of the initial differential equation assists to solve the obtained algebraic system and, thereby, simplifies the search of elliptic solutions. This procedure has been automatized and generalized on some type of multivalued solutions. To find solutions of the initial differential equation in the form of the Laurent or Puiseux series we use the Painlevé test. This test can also assist to solve the inverse problem: to find the form of a polynomial potential, which corresponds to the required type of solutions. We consider the five–dimensional gravitational model with a scalar field to demonstrate it.