We characterize the Liouvillian and analytic integrability of the quadratic polynomial vector fields in ℝ2 having an invariant ellipse. More precisely, a quadratic system having an invariant ellipse can be written into the form ẋ = x2 + y2 - 1 + y (ax + by + c}, ẏ = - x(ax + by + c}, and the ellipse becomes x 2 + y 2 = 1. We prove that (i) this quadratic system is analytic integrable if and only if a = 0 (ii) if x 2+y 2 = 1 is a periodic orbit, then this quadratic system is Liouvillian integrable if and only if x 2 + y 2 = 1 is not a limit cycle; and (iii) if x2 +y 2 = 1 is not a periodic orbit, then this quadratic system is Liouvilian integrable if and only if a = 0. © 2014 Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg.
- invariant ellipse
- Liouvillian integrability
- quadratic planar polynomial vector fields