Recently, the authors provided an example of an integrable Liouvillian planar polynomial differential system that has no finite invariant algebraic curves; see Giné and Llibre (2012). In this note, we prove that, if a complex differential equation of the form y '=a 0(x)+a 1(x)y+⋯+a n(x)y n, with a i(x) polynomials for i=0, 1, . . ., n, a n(x)≠0, and n≥2, has a Liouvillian first integral, then it has a finite invariant algebraic curve. So, this result applies to Riccati and Abel polynomial differential equations. We shall prove that in general this result is not true when n=1, i.e., for linear polynomial differential equations. © 2012 Elsevier Ltd.
- Abel differential equation
- Invariant algebraic curve
- Liouvillian integrability
- Riccati differential equation