Darboux Points and Integrability of Homogeneous Hamiltonian Systems with Three and More Degrees of Freedom. Nongeneric Cases

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 $V'(\boldsymbol{d})=0$. The existence of such a solution gives very strong integrability obstructions obtained in the frame of the Morales–Ramis theory. This theory also gives additional integrability obstructions which have the form of restrictions imposed on the eigenvalues $(\lambda_1,\ldots,\lambda_n)$ of the Hessian matrix $V''(\boldsymbol{d})$ calculated at a nonzero $\boldsymbol{d}\in\mathbb{C}^n$ satisfying $V'(\boldsymbol{d})=\boldsymbol{d}$. In our previous work we showed that for generic potentials some universal relations between $(\lambda_1,\ldots,\lambda_{n})$ calculated at various solutions of $V'(\boldsymbol{d})=\boldsymbol{d}$ exist. These relations allow one to prove that the number of potentials satisfying the necessary conditions for the integrability is finite. The main aim of this paper was to show that relations of such forms also exist for nongeneric potentials. We show their existence and derive them for the case $n=k=3$ applying the multivariable residue calculus. We demonstrate the strength of the results analyzing in details the nongeneric cases for $n=k=3$. Our analysis covers 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 superintegrable with additional polynomial first integrals which seemingly can be of an arbitrarily high degree with respect to the momenta was distinguished.