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

We study the integrability in the Liouville sense of natural Hamiltonian systems with a homogeneous rational potential $V(\vq)$. Strong necessary conditions for the integrability of such systems were obtained by an analysis of differential Galois group of variational equations along certain particular solutions. These conditions have the form of arithmetic restrictions putted on eigenvalues of Hessian $V"(\vd)$ calculated at a non-zero solution $\vd$ of equation $\grad V(\vd)=\vd$. Such solutions are called proper Darboux points. It was recently proved that for generic polynomial homogeneous potentials there exist universal relations between eigenvalues of Hessians of the potential taken at all proper Darboux points. The question about the existence of such relations for rational potentials seems to be hard. One of the reason of this fact 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 admissible values for eigenvalues of Hessian at a proper Darboux point for potentials satisfying necessary conditions for the integrability is finite. In turn, it gives a tool for classification of integrable rational potentials. Also, quite recently, it was shoved that for polynomial homogeneous potentials additional necessary conditions for the integrability can be deduced from the existence of improper Darboux points, i.e., points $\vd$ which are non-zero solution of equation $\grad V(\vd)=\vzero$. These new conditions have also the form of arithmetic restrictions imposed on eigenvalues of $V"(\vd)$. In this paper we prove that for rational potentials improper Darboux points give the same necessary conditions for the integrability.