Resumen de Global C∞ Integrability of Cubic–Linear Polynomial Differential Systems

Yangyou Pan, Cong Wang, Xiang Zhang

• The cubic–linear polynomial differential systems having at least one finite singularity are affine equivalent to the systems of the form x = P(x, y) = bx + cy + dx2 + exy + f y2 + gx3 + hx2 y + ixy2 + jy3, y = Q(x, y), with g2 + h2 + i 2 + j 2 = 0 (otherwise it is quadratic–linear), and Q(x, y) is either x or y. In this paper we classify all the cubic–linear systems with Q(x, y) = y which have a global C∞ first integral. Meanwhile we obtain some partial results on the existence of global analytic first integrals. For proving our results we will use the local characterization of first integrals, partition of unity in R2, smoothness of first integrals in canonical regions and so on.