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The existence of quasi-bi-Hamiltonian structures for a two-dimensionalsuperintegrable $(k_1,k_2,k_3)$-dependent Kepler-related problem is studied. Wemake use of an approach that is related with the existence of some complexfunctions which satisfy interesting Poisson bracket relations and that waspreviously applied to the standard Kepler problem as well as to some particularsuperintegrable systems as the Smorodinsky-Winternitz (SW) system, theTremblay-Turbiner-Winternitz (TTW) and Post-Winternitz (PW) systems. We provethat these complex functions are important for two reasons: first, theydetermine the integrals of motion, and second they determine the existence ofsome geometric structures (in this particular case, quasi-bi-Hamiltonianstructures).  All the results depend of three parameters ($k_1, k_2, k_3$) in such a waythat in the particular case $k_1\ne 0$, $k_2= k_3= 0$, we recover the resultsof the original Kepler problem (previously studied in SIGMA 12, 010 (2016)).This paper can be considered as divided in two parts and every part present adifferent approach (different complex functions and differentquasi-bi-Hamiltonian structures).

Complex functions and geometric structures associated to the  superintegrable Kepler-related family of systems endowed with generalized  Runge-Lenz integrals of motion