Polynomial and rational first integrals for planar quasi-homogeneous polynomial differential systems (Q379592)

The authors characterize those quasi-homogeneous polynomial systems of ordinary differential equations on the plane that admit either a polynomial or a rational first integral. A key result of independent interest is that any quasi-homogeneous polynomial system of ordinary differential equations on the plane can be transformed into a system of the form NEWLINE\[NEWLINE\dot u = u f(v),\, \dot v = g(v)NEWLINE\]NEWLINE for some pair of relatively prime polynomials \(f\) and \(g\).