Alexey TSYGVINTSEV

Section of theoretical mechanics
Departament of Mathematics and Mechanics
Moskow State University
Vorob'evy Gory, 119899, Moskow, Russia
tsygvin@nw.math.msu.su

On the algebraic integrability of one class of differential systems

Abstract: We consider systems of ordinary differential equations with quadratic homogeneous right hand side. Systems of such a form arise in many problems of classical mechanics: Euler-Poincare equations on Lie algebras, the Lotka-Volterra systems,etc. The main concern of this work is to find the cases then these equations have first integrals. Using the generalized Lagutinski method we find necessary conditions for the existence of polynomial first integrals and symmetry fields. We proof that every homogeneous first integral is a linear combination of a fixed set of polynomials called base functions. As an example the case of two equations is considered. The classification of possible polynomial first integrals is done.