Nijenhuis tensor and invariant polynomials

We discuss the diagonalization problem of Nijenhuis tensor in a class Poisson-Nijenhuis structures defined on compact hermitian symmetric spaces. study its action ring invariant polynomials Thimm chain subalgebras. The existence $\phi$-minimal representations defines suitable basis that completely solves problem. prove such exist classical cases AIII, BDI, DIII and CI, do not exceptional EIII EVII. second general construction these two computes partially spectrum hints at different behavior with respect to cases.