Qualitative study of a model with Rastall gravity

We consider the Rastall theory for the flat Friedmann–Robertson–Walker Universe filled with a perfect fluid that satisfies a linear equation of state. The corresponding dynamical system is a two dimensional system of polynomial differential equations depending on four parameters. We show that this differential system is always Darboux integrable. In order to study the global dynamics of this family of differential systems we classify all their non-topological equivalent phase portraits in the Poincare disc and we obtain 16 different dynamical situations for our spacetime.