Institute of Nuclear Physics,

Moscow State University,

Vorobevy Gory,

119992 Moscow,

RUSSIA

E-mail: svernov@theory.sinp.msu.ru

**From the Laurent-series solutions to elliptic solutions
for dynamical systems**

**Abstract:**

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.