Polynomial differential systems having a given Darbouxian first integral

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 1 λ 1 ⋯ f p λ p exp( g/ h) where f i , g and h are polynomials in C[x,y] , and λ i∈ C for i=1,…, p. The functions of this form are called Darbouxian functions. 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.