On Liouvillian integrability of the first–order polynomial ordinary differential equations

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) [8]. In this note, we prove that, if a complex differential equation of the form y′=a0(x)+a1(x)y+⋯+an(x)yny′=a0(x)+a1(x)y+⋯+an(x)yn, with ai(x)ai(x) polynomials for i=0,1,…,ni=0,1,…,n, an(x)≠0an(x)≠0, and n≥2n≥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=1n=1, i.e., for linear polynomial differential equations.