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

中文翻译：

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准双哈密顿结构和超可积性：研究具有广义 Runge-Lenz 运动积分的开普勒相关系统族

研究了二维超可积$(k_1,k_2,k_3)$依赖开普勒相关问题的拟双哈密顿结构的存在性。我们使用了一种方法，该方法与一些满足有趣的泊松括号关系的复杂函数的存在相关，该方法以前应用于标准开普勒问题以及一些特定的超可积系统，如 Smorodinsky-Winternitz (SW) 系统， Tremblay-Turbiner-Winternitz (TTW) 和 Post-Winternitz (PW) 系统。我们证明这些复函数很重要，原因有二：首先，它们决定了运动的积分，其次，它们决定了某些几何结构（在这种特殊情况下，准双哈密顿结构）的存在。所有的结果都取决于三个参数（$k_1、k_2、