Superintegrability of Generalized Toda Models on Symmetric Spaces

In this paper we prove superintegrability of Hamiltonian systems generated by functions on $K\mathrm{\backslash }G\mathrm{/}K$, restriced to a symplectic leaf of the Poisson variety $G\mathrm{/}K$, where $G$ is a simple Lie group with the standard Poisson Lie structure, $K$ is the subgroup of fixed points with respect to the Cartan involution.