Poisson-commutative subalgebras and complete integrability on non-regular coadjoint orbits and flag varieties

The purpose of this paper is to bring together various loose ends in the theory of integrable systems. For a semisimple Lie algebra $\mathfrak{g}$, we obtain several results on completeness of homogeneous Poisson-commutative subalgebras of $\mathcal{S}(\mathfrak{g})$ on coadjoint orbits. This concerns, in particular, Mishchenko-Fomenko and Gelfand-Tsetlin subalgebras.