In previous work, we have considered Hamiltonians associated with three-dimensional conformally flat spaces, possessing two-, three- and four-dimensional isometry algebras. Previously, our Hamiltonians have represented free motion, but here we consider the problem of adding potential functions in the presence of symmetry. Separable potentials in the three-dimensional space reduce to 3 or 4 parameter potentials for Darboux–Koenigs Hamiltonians. Other three-dimensional coordinate systems reveal connections between Darboux–Koenigs and other well-known super-integrable Hamiltonians, such as the Kepler problem and isotropic oscillator.

中文翻译：

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为具有对称性的超可积系统增加势能

在先前的工作中，我们已经考虑了与三维共形平面空间相关的哈密顿量，该空间具有二维，三维和四维等距代数。以前，我们的哈密顿量表示自由运动，但在这里我们考虑在对称的情况下添加潜在函数的问题。对于Darboux–Koenigs哈密顿量，三维空间中的可分势降为3或4个参数势。其他三维坐标系则揭示了Darboux–Koenigs与其他著名的超可积分哈密顿量之间的联系，例如开普勒问题和各向同性振荡器。