We solve the classical and quantum problems for the 1D sigma model with target space the flag manifold \(\textrm{U}(3)\over \textrm{U}(1)^3\) , equipped with the most general invariant metric. In particular, we explicitly describe all geodesics in terms of elliptic functions and demonstrate that the spectrum of the Laplace–Beltrami operator may be found by solving polynomial (Bethe) equations. The main technical tool that we use is a mapping between the sigma model and a Gaudin model, which is also shown to hold in the \(\textrm{U}(n)\) case.
