A Note about Integrable Systems on Low-dimensional Lie Groups and Lie Algebras

Alexey Bolsinov*, Jinrong Bao

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

5 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)266-280
Number of pages15
JournalRegular and Chaotic Dynamics
Issue number3
Publication statusPublished - 1 May 2019
Externally publishedYes


  • 37J35
  • 53B50
  • 70H06
  • 70S10
  • Integrable systems
  • Lie groups
  • geodesic flow
  • left-invariant metric
  • sub-Riemannian structure

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