  We study a class of Poisson-Nijenhuis systems defined on compact hermitian
symmetric spaces, where the Nijenhuis tensor is defined as the composition of
Kirillov-Konstant-Souriau symplectic form with the so called Bruhat-Poisson
structure. We determine its spectrum. In the case of Grassmannians the
eigenvalues are the Gelfand-Tsetlin variables. We introduce the abelian algebra
of collective hamiltonians defined by a chain of nested subalgebras and prove
complete integrability. By construction, these models are integrable with
respect to both Poisson structures. The eigenvalues of the Nijenhuis tensor are
a choice of action variables. Our proof relies on an explicit formula for the
contravariant connection defined on vector bundles that are Poisson with
respect to the Bruhat-Poisson structure.
