The Darbouxian theory of integrability allows to determine when a polynomial differential system in ℂ2 has a first integral of the kind f1λ1... fpλpexp(g/h) where fi, g and h are polynomials in ℂ[x,y], and λi ∈ 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 ℂ2 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. © 2004 Elsevier SAS. All rights reserved.
- Darbouxian function
- First integral
- Polynomial differential system