Rational integrability of two-dimensional quasi-homogeneous polynomial differential systems (Q984100)

The authors prove three theorems that characterize in various ways when a quasi-homogeneous polynomial vector field on the plane (or the system of first order differential equations that corresponds to it) admits a rational first integral. The characterizations are stated in terms of the decomposition that is known to exist of such a vector field into a sum of a conservative and a dissipative vector field. One characterization relates rational integrability to the Kowalevskaya exponents of the vector field. They illustrate their methods with an application to \((1,2)\)-polynomial systems of degree two.