Isotropic embeddings of coadjoint orbits and magnetic geodesic flows

We consider isotropic and Lagrangian embeddings of coadjoint orbits of compact Lie groups into products of coadjoint orbits. After reviewing the known facts in the case of $\mathrm{S}\mathrm{U}(n)$ we initiate a similar study for $\mathrm{S}\mathrm{O}$ and $\mathrm{S}\mathrm{p}$ cases. In the second part we apply this to the study of dynamical systems with $\mathrm{S}\mathrm{U}(n)$ symmetry, proving equivalence between systems of two types: those describing magnetic geodesic flow on flag manifolds and classical `spin chains’ of a special type.