Abstract:

We use tecnics of Differential Galois theory in order to study the integrability of polynomial vector fields when their associated foliation is reduced to a Riccati. More concrete we study some families of quadratic vector fields, Lienard equations and equations related with special functions such as Hypergeometric and Heun ones. Additionally, we consider the Poincaré problem in some of the families.
Kovacic's Algorithm is a powerfull tool in this study.