Geometry of translations of invariants on semisimple Lie algebras

Each orbit of the coadjoint representation of a semisimple Lie algebra can be equipped with a complete commutative family of polynomials; this family was obtained by the argument-translation method in papers of Mishchenko and Fomenko. This commutative family and the corresponding Euler's equations play an important role in the theory of finite-dimensional integrable systems. These Euler's equations admit a natural Lax representation with spectral parameter.
It is proved in the paper that the discriminant of the spectral curve coincides completely with the bifurcation diagram of the moment map for the algebra $\mathrm{sl}(n,\mathbb C)$. The maximal degeneracy points of the moment map are described for compact semisimple Lie algebras in terms of the root structure. It is also proved that the set of regular points of the moment map is connected, and the inverse image of each regular point consists of precisely one Liouville torus.