This talk deals with a class of left-invariant geodesic flows over real semi-simple Lie groups introduced by Mishchenko and Fomenko around 1980. Using the bi-Hamiltonian method proposed by Bolsinov, Oshemkov, and Izosimov, the complete integrability of the associated Lie-Poisson equation is explained. Then, the Williamson types of the isolated equilibria on generic adjoint orbits are classified in terms of root systems. Related dynamical systems are also discussed. A class of systems arising from statistical transformation models will be mentioned.

The talks is based on joint works with Tudor S. Ratiu (Shanghai Jiao Tong University) and with Jean-Pierre Françoise (LJLL, Sorbonne Université).

Financed by FCT – Fundação para a Ciência e a Tecnologia, I.P., ref. UIDB/00208/2020.