Inverse problems for multiple invariant curves

Planar polynomial vector fields which admit invariant algebraic curves, Darboux integrating factors or Darboux first integrals axe of special interest. We solve the inverse problem for invariant algebraic curves with a given multiplicity and for integrating factors, under generic assumptions regarding the (multiple) invariant algebraic curves involved. In particular, we prove, in this generic scenario, that the existence of a Darboux integrating factor implies Darboux integrability. Furthermore, we construct examples in which the genericity assumption does not hold and indicate that the situation is different for these. © 2007 The Royal Society of Edinburgh.