On the Rational Integrals of Two-Dimensional Natural Systems

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