Polynomial differential systems having a given Darbouxian first integral*1

Abstract

The Darbouxian theory of integrability allows to determine when a polynomial differential system in C 2 has a first integral of the kind f Here, we solve the inverse problem, i.e. we characterize the polynomial vector fields in C 2 having a given Darbouxian function as a first integral.On the other hand, using information about the degree of the invariant algebraic curves of a polynomial vector field, we improve the conditions for the existence of an integrating factor in the Darbouxian theory of integrability.