Quadratic systems with an integrable saddle: A complete classification in the coefficient space R12

A quadratic polynomial differential system can be identified with a single point of R12 through the coefficients. Using the algebraic invariant theory we classify all the quadratic polynomial differential systems of R12 having an integrable saddle. We show that there are only 47 topologically different phase portraits in the Poincaré disk associated to this family of quadratic systems up to a reversal of the sense of their orbits. Moreover each one of these 47 representatives is determined by a set of affine invariant conditions.