On the Generalized Hartmann Model for the Benzene Molecule

In this work we study a generalized integrable biparametric family of 4-D isotropic oscillators. This family allows to treat, in a uni ed way, oscillators de ned by the potential given by Hartmann related with the benzene molecule and other ring shape ones. These potentials belong to two larger families of 3-D integrable Hamiltonian systems and have received a special attention since are related with the benzene molecule, as well as other models in quantum chemistry and nuclear physics. Using the Liouville-Arnold theorem and a particular analysis of the momentum map at its critical points, a complete topological classi cation of the di erent invariant sets of the phase ow of this problem is presented. By means of this topological study and the calculation of the action-angle variables the classi cation of periodic and quasiperiodic orbits are stated. The main aim of this paper is to study the stability properties of these orbits and to prove the integrability of the system correcting a mathematical mistake of [7].