We consider Hamiltonians associated with 3 dimensional conformally flat spaces, possessing 2, 3 and 4 dimensional isometry algebras. We use the conformal algebra to build additional quadratic first integrals, thus constructing a large class of superintegrable systems and the complete Poisson algebra of first integrals. We then use the isometries to reduce our systems to 2 degrees of freedom. For each isometry algebra we give a universal reduction of the corresponding general Hamiltonian. The superintegrable specialisations reduce, in this way, to systems of Darboux–Koenigs type, whose integrals are reductions of those of the 3 dimensional system.

中文翻译：

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三维共形平面空间上的超可积系统

我们考虑与 3 维共形平坦空间相关的哈密顿量，具有 2、3 和 4 维等距代数。我们使用保形代数来构建额外的二次一阶积分，从而构建了一大类超可积系统和一阶积分的完全泊松代数。然后我们使用等距将我们的系统减少到 2 个自由度。对于每个等距代数，我们给出相应的一般哈密顿量的普遍约简。以这种方式，超可积特化归约为 Darboux-Koenigs 类型的系统，其积分是 3 维系统的归约。