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Global \(C^\infty\) integrability of quartic-linear polynomial differential systems. (English) Zbl 1411.34044

Summary: The quartic-linear polynomial differential systems having at least one finite singularity are affine equivalent to systems of the form \[\begin{split}\dot{x}&=P(x,y)= P_1(x,y)+P_2(x,y) +P_3(x,y) +P_4(x,y), \\ \dot{y} &=Q(x,y)\end{split}\] where \(P\) and \(Q\) are coprime, \(P_i\) are homogeneous polynomials of degree \(i\) and \(p_4\not\equiv 0\) (otherwise it is cubic-linear) and \(Q(x,y)\) is either \(x\) or \(y\). In this paper, we classify all the quartic-linear systems with \(Q(x,y)=y\) which have a global \(C^\infty\) first integral. We use the local characterization of first integrals, partition of unity in \(\mathbb{R}^2\) and smoothness of first integrals in canonical regions.

MSC:

34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
34C23 Bifurcation theory for ordinary differential equations
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