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 ′ = 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.