Phase portraits and invariant straight lines of cubic polynomial vector fields having a quadratic rational first integral

In this paper we classify all cubic polynomial differential systems having a rational first integral of degree two. In other words we characterize all the global phase portraits of the cubic polynomial differential systems having all their orbits contained in conies. We also determine their configurations of invariant straight lines. We show that there are exactly 38 topologically different phase portraits in the Poincare' disc associated with this family of cubic polynomial differential systems up to a reversed sense of their orbits. Copyright © 2011 Rocky Mountain Mathematics Consortium.