Darboux points and integrability of homogeneous Hamiltonian systems with three and more degrees of freedom. Nongeneric cases

Abstract

In this paper the problem of classification of integrable natural Hamiltonian systems with $n$ degrees of freedom given by a Hamilton function which is the sum of the standard kinetic energy and a homogeneous polynomial potential $V$ of degree $k>2$ is investigated. It is assumed that the potential is not generic. Except for some particular cases a potential $V$ is not generic, if it admits a nonzero solution of equation $V'(\vd)=0$. The existence of such solution gives very strong integrability obstructions obtained in the frame of the Morales-Ramis theory. This theory gives also additional integrability obstructions which have the form of restrictions imposed on the eigenvalues $(\lambda_1,...,\lambda_n)$ of the Hessian matrix $V''(\vd)$ calculated at a non-zero $\vd\in\C^n$ satisfying $V'(\vd)=\vd$. Furthermore, we show that similarly to the generic case also for nongeneric potentials some universal relations between $(\lambda_1,...,\lambda_{n})$ calculated at various solutions of $V'(\vd)=\vd$ exist. We derive them for case $n=k=3$ applying the multivariable residue calculus. We demonstrate the strength of the obtained results analysing in details the nongeneric cases for $n=k=3$. Our analysis cover all the possibilities and we distinguish those cases where known methods are too weak to decide if the potential is integrable or not. Moreover, for $n=k=3$ thanks to this analysis a three-parameter family of potentials integrable or super-integrable with additional polynomial first integrals which seemingly can be of an arbitrarily high degree with respect to the momenta was distinguished.