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We study the integrability of polynomial vector fields using Galois theory oflinear differential equations when the associated foliations is reduced to aRiccati type foliation. In particular we obtain integrability results for somefamilies of quadratic vector fields, Li\'enard equations and equations relatedwith special functions such as Hypergeometric and Heun ones. The Poincar\'eproblem for some families is also approached.

On the integrability of polynomial fields in the plane by means of Picard-Vessiot theory