Almost multiplicity free subgroups of compact Lie groups and polynomial integrability of sub-Riemannian geodesic flows

We classify almost multiplicity free subgroups K of compact simple Lie groups G. The problem is related to the integrability of Riemannian and sub-Riemannian geodesic flows of left-invariant metrics defined by a specific extension of integrable systems from T∗K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T^*K$$\end{document} to T∗G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T^*G$$\end{document}.