Quasi-bi-Hamiltonian structures and superintegrability: Study of a Kepler-related family of systems endowed with generalized Runge-Lenz integrals of motion

Abstract

The existence of quasi-bi-Hamiltonian structures for a two-dimen-sional superintegrable \begin{document}$ (k_1,k_2,k_3) $\end{document} -dependent Kepler-related problem is studied. We make use of an approach that is related with the existence of some complex functions which satisfy interesting Poisson bracket relations and that was previously applied to the standard Kepler problem as well as to some particular superintegrable systems as the Smorodinsky-Winternitz (SW) system, the Tremblay-Turbiner-Winternitz (TTW) and Post-Winternitz (PW) systems. We prove that these complex functions are important for two reasons: first, they determine the integrals of motion, and second they determine the existence of some geometric structures (in this particular case, quasi-bi-Hamiltonian structures). All the results depend on three parameters ( \begin{document}$ k_1, k_2, k_3 $\end{document} ) in such a way that in the particular case \begin{document}$ k_1\ne 0 $\end{document} , \begin{document}$ k_2 = k_3 = 0 $\end{document} , the properties characterizing the Kepler problem are obtained. This paper can be considered as divided in two parts and every part presents a different approach (different complex functions and different quasi-bi-Hamil-tonian structures).