  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.
