Algebraically integrable quadratic dynamical systems

Abstract: We consider in C^n the class of symmetric homogeneous quadratic dynamical systems. We introduce the notion of algebraic integrability for this class. We present a class of symmetric quadratic dynamical systems that are algebraically integrable by the set of functions h_1(t), ..., h_n(t) where h_1(t) is any solution of an ordinary differential equation of order n and h_k(t) are differential polynomials in h_1(t), k = 2, ..., n. We describe a method of constructing this ordinary differential equation. We give a classification of symmetric quadratic dynamical systems and describe the maximal subgroup in GL(n, C) that acts on this systems. We apply our results to analysis of classical systems of Lotka-Volterra type and Darboux-Halphen system and their modern generalizations.