Phase portraits of the quadratic vector fields with a polynomial first integral

In this work we classify the phase portraits of all quadratic polynomial differential systems having a polynomial first integral. IfH(x, y) is a polynomial of degreen+1 then the differential system is called a Hamiltonian system of degree n. We also prove that all the phase portraits that we obtain in this paper are realizable by Hamiltonian systems of degree 2. Since we observe that all the phase portraits of the linear polynomial differential systems having a polynomial first integral are also realizable by Hamiltonian systems of degree 1, an open question appears: Are all the phase portraits of polynomial differential systems of degree n having a polynomial first integral realizable by Hamiltonian systems of degree n? © 2006 Springer.