Affine transformations in Euclidean space generate a correspondence between integrable systems on cotangent bundles to a sphere, ellipsoid and hyperboloid embedded in \(R^{n}\). Using this correspondence and the suitable coupling constant transformations, we can get real integrals of motion in the hyperboloid case starting with real integrals of motion in the sphere case. We discuss a few such integrable systems with invariants which are cubic, quartic and sextic polynomials in momenta.

中文翻译：

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球体、椭球体和双曲面上的可积系统

欧几里得空间中的仿射变换生成余切束上的可积系统与嵌入\(R^{n}\)中的球体、椭球体和双曲面之间的对应关系。使用这种对应关系和适当的耦合常数变换，我们可以从球体情况下的运动实积分开始获得双曲面情况下的运动实积分。我们讨论了一些这样的可积系统，其不变量是动量的三次、四次和六次多项式。