### Abstract

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.

Original language | English |
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Pages (from-to) | 775-788 |

Journal | Bulletin des Sciences Mathematiques |

Volume | 128 |

DOIs | |

Publication status | Published - 1 Oct 2004 |

### Keywords

- Darbouxian function
- First integral
- Polynomial differential system

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## Cite this

Llibre, J., & Pantazi, C. (2004). Polynomial differential systems having a given Darbouxian first integral.

*Bulletin des Sciences Mathematiques*,*128*, 775-788. https://doi.org/10.1016/j.bulsci.2004.04.001