SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 2, Page 1525-1539, January 2020.
This work deals with the full reduction of the spatial Kepler system for bounded and unbounded motions. Precisely, we consider the four-dimensional oscillator associated to the Kepler system and carry out our program in three stages: axial-axial-energy rather than energy-axial-axial as is customary. This approach reveals the true role of the $\mathcal{KS}$ map and the bilinear relation. Our development allows for a global analysis at each reduction stage providing a complete description of each reduced space. The first reduced space is described by means of only six invariants leading to a nonsingular six-dimensional Poisson manifold. Then, the classical bilinear relation is replaced by the momentum map of a geometric reduction. Furthermore, the second reduced space is given as the product of two hyperboloids and has cones as singular strata. For the last stage, we distinguish among the possible sign of the energy. The positive and zero cases bring new noncompact reduced spaces which are described in detail.

中文翻译：

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开普勒系统各阶段的替代约简。正，负和零能量

SIAM应用动力系统杂志，第19卷，第2期，第1525-1539页，2020年1月。
这项工作涉及完全消除有界和无界运动的空间开普勒系统。准确地说，我们考虑与开普勒系统相关的四维振荡器，并按三个阶段执行程序：轴向-轴向能量，而不是通常的轴向能量。这种方法揭示了$ \ mathcal {KS} $映射和双线性关系的真实作用。我们的开发允许在每个缩减阶段进行全局分析，从而提供每个缩减空间的完整描述。仅通过导致一个非奇异的六维泊松流形的六个不变量来描述第一个缩小的空间。然后，将经典的双线性关系替换为几何简化的动量图。此外，第二个缩小的空间是两个双曲面的乘积，并且视锥为奇异层。在最后阶段，我们区分能量的可能迹象。正和零的情况带来了新的非紧缩空间，将对此进行详细描述。