Polynomial integrability of sub-Riemannian geodesic flows on compact Lie groups

We classify almost multiplicity-free subgroups K of compact simple Lie groups G. This problem is connected to the integrability of Riemannian and sub-Riemannian geodesic flows of left-invariant metrics, which are defined by a specific extension of integrable systems from T^*K to T^*G.
To construct integrable nonholonomic and sub-Riemannian flows with left-invariant metrics and nonholonomic distributions on compact Lie groups, we use chains of subalgebras. While the nonholonomic problem is not Hamiltonian, the sub-Riemannian problem is.
This talk is based on the paper Almost Multiplicity-Free Subgroups of Compact Lie Groups and Polynomial Integrability of Sub-Riemannian Geodesic Flows by Božidar Jovanović, Tijana Šukilović, and Srdjan Vukmirović [Letters in Mathematical Physics, 114(1), Article no. 14, 2024].