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We study the integrability in the Liouville sense of natural Hamiltoniansystems with a homogeneous rational potential $V(\vq)$. Strong necessaryconditions for the integrability of such systems were obtained by an analysisof differential Galois group of variational equations along certain particularsolutions. These conditions have the form of arithmetic restrictions putted oneigenvalues of Hessian $V"(\vd)$ calculated at a non-zero solution $\vd$ ofequation $\grad V(\vd)=\vd$. Such solutions are called proper Darboux points.It was recently proved that for generic polynomial homogeneous potentials thereexist universal relations between eigenvalues of Hessians of the potentialtaken at all proper Darboux points. The question about the existence of suchrelations for rational potentials seems to be hard. One of the reason of thisfact is the presence of indeterminacy points of the potential and its gradient.Nevertheless, for two degrees of freedom we prove that such relation exists.This result is important because it allows to show that the set of admissiblevalues for eigenvalues of Hessian at a proper Darboux point for potentialssatisfying necessary conditions for the integrability is finite. In turn, itgives a tool for classification of integrable rational potentials. Also, quiterecently, it was shoved that for polynomial homogeneous potentials additionalnecessary conditions for the integrability can be deduced from the existence ofimproper Darboux points, i.e., points $\vd$ which are non-zero solution ofequation $\grad V(\vd)=\vzero$. These new conditions have also the form ofarithmetic restrictions imposed on eigenvalues of $V"(\vd)$. In this paper weprove that for rational potentials improper Darboux points give the samenecessary conditions for the integrability.

Darboux Points and Integrability Analysis of Hamiltonian Systems with Homogeneous Rational Potentials