On integrability of transverse Lie–Poisson structures at nilpotent elements

We construct families of functions in involution for transverse Poisson structures at nilpotent elements of Lie–Poisson structures on simple Lie algebras by using the argument shift method. Examples show that these families contain completely integrable systems that consist of polynomial functions. We provide a uniform construction of these integrable systems for an infinite family of distinguished nilpotent elements of semisimple type.