  Polynomial dynamical systems describing interacting particles in the plane
are studied. A method replacing integration of a polynomial multi--particle
dynamical system by finding polynomial solutions of a partial differential
equations is described. The method enables one to integrate a wide class of
polynomial multi--particle dynamical systems. The general solutions of certain
dynamical systems related to linear second--order partial differential
equations are found. As a by-product of our results, new families of orthogonal
polynomials are derived. Our approach is also applicable to dynamical systems
that are not multi--particle by their nature but that can be regarded as
multi--particle (for example, the Darboux--Halphen system and its
generalizations). A wide class of two and three--particle polynomial dynamical
systems is integrated.
