From the Laurent-series solutions to elliptic solutions for dynamical systems
The Painleve test is very useful to construct not only the Laurent-series solutions, but also solutions in terms of the elliptic or elementary functions. All such functions are solutions of the first order differential equations. The direct algebraic method is the substitution of the corresponding first order equation in the initial differential equation to transform it in a nonlinear algebraic system on coefficients of the first order equation and parameters of the initial differential equation. It can be too difficult to solve the obtained system by the Groebner basis method. The use of the Laurent series solutions gives additional algebraic equations, which are linear in coefficients of the first order equation and nonlinear, maybe even nonpolynomial, in parameters of the initial equation. The additional equations are not consequences of the initial algebraic system, so some parameters should be fixed. In contrast to the Groebner basis method, this method allows to find some solutions of the algebraic system, which can not be solved. The algorithm for construction of the additional algebraic equations has been implemented in the computer algebra systems Maple and REDUCE. It is possible to find the analytic form of some multivalued solutions as well.