Quadratic systems with a polynomial first integral: A complete classification in the coefficient space R12

In this paper we are going to apply the invariant theory to give invariant conditions on the coefficients of any non-degenerate quadratic system in order to determine if it has or not a polynomial first integral without using any normal form. We obtain that the existence of polynomial first integral is directly related with the fact that all the roots of a convenient cubic polynomial are rational and negative. The coefficients of this cubic polynomial are invariants related with some geometric properties of the system.