Jesús S. Pérez del Río

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 systemx′=−∂H/∂y,y′=∂H/∂x is called a Hamiltonian system of degreen. 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 degreen having a polynomial first integral realizable by Hamiltonian systems of degreen?

Planar polynomial vector fields having a polynomial first integral can be obtained from linear systems

We consider in this work planar polynomial differential systems having a polynomial first integral. We prove that these systems can be obtained from a linear system through a polynomial transformation of variables.