On the rational integrals of two-dimensional natural systems

We study a natural mechanical system having an additional first integral in the form of a function rational in momenta. One of the authors has proved recently that if the configuration space of the system is the two-dimensional torus; then, provided that the potential is analytic, the existence of a rational integral with analytic periodic coefficients and small degrees of the numerator and denominator implies the existence of an integral linear in momenta. In the present article, this result is generalized to the case that the configuration space of the system is the two-dimensional plane.