A criterion for existence of rational general solutions of planar systems of ODEs

In the paper [Ngo09] we have studied the algebraic ODE of first order F (x, y, y′) = 0, where F ∈ Q[x, y, z], given its proper rational parametrization of the corresponding surface F (x, y, z) = 0. Using this proper parametrization we deduced the problem of finding rational general solutions of the equation F (x, y, y′) = 0 to finding rational general solutions of its associated system of ODEs in two new indeterminates s, t. This is a planar autonomous system of first order in s, t and of first degree in s′, t′. In this paper we give a criterion for existence of rational general solutions of such an autonomous system provided a degree bound of its rational general solutions. The criterion is based on the vanishing of the differential pseudo remainder of Gao’s differential polynomials [FG06] with respect to the chain of the ODE system. As a result, we use this criterion to classify all planar linear systems of ODEs having a rational general solution.