Bi-homogeneity and integrability of rational potentials

In this paper we consider natural Hamiltonian systems with two degrees of freedom for which Hamiltonian function has the form H = 1/2 (p(1)(2) + p(2)(2)) + V (q(1), q(2)) and potential V (q(1), q(2)) is a rational function. Necessary conditions for the integrability of such systems are deduced from integrability of dominate term of the potential which usually is appropriately chosen homogeneous term of V. We show that introducing weights compatible with the canonical structure one can find new dominant terms which can give new necessary conditions for integrability. To deduce them we investigate integrability of a family of bi-homogeneous potentials which depend on two integer parameters. Unexpectedly systems with these potentials can be reduced to the Lotka-Volterra quadratic planar vector field. Then theorem of Jean Moulin Oilagnier, allows us to make complete classification of integrable cases. Moreover, the reduction is used for explicit integration of an exceptional case with a polynomial first integral of degree 4 in momenta. (C) 2019 Elsevier Inc. All rights reserved.