The goal of the paper is to explain why any left-invariant Hamiltonian system on (the cotangent bundle of) a 3-dimensonal Lie group G is Liouville integrable. We derive this property from the fact that the coadjoint orbits of G are two-dimensional so that the integrability of left-invariant systems is a common property of all such groups regardless their dimension.We also give normal forms for left-invariant Riemannian and sub-Riemannian metrics on 3-dimensional Lie groups focusing on the case of solvable groups, as the cases of SO(3) and SL(2) have been already extensively studied. Our description is explicit and is given in global coordinates on G which allows one to easily obtain parametric equations of geodesics in quadratures.

中文翻译：

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关于低维李群和李代数的可积系统的一个注记

本文的目的是解释为什么3维李氏群G的切余束上的任何左不变哈密顿系统都是Liouville可积的。我们从G的共轭轨道是二维的这一事实得出这一性质，因此左不变系统的可积性是所有此类群的一个共同特性，而不论它们的维数如何。我们还给出了左不变黎曼和子方程的范式-关于SO（3）和SL（2）的情况的3维Lie组的黎曼度量重点放在可解组的情况上。我们的描述是明确的，并以G上的全局坐标给出 这使人们可以轻松地求出大地测量学的参数方程式。