Polynomial first integrals of quadratic systems

The main purpose of this paper is to give the classification and the topological phase portraits of all quadratic systems having minimal polynomial first integrals of degree less than 5, and to prove the existence of minimal polynomial first integrals of any degree for quadratic systems. Moreover, we prove that quadratic systems with minimal polynomial first integrals of degree larger than 1 have at most three invariant straight lines, and under convenient assumptions we give the greatest degree of the irreducible polynomial firstin tegrals. © 2001 Rocky Mountain Mathematics Consortium.