Perfect observables for the hierarchical non-linear O(N)-invariant σ-model

We compute moving eigenvalues and the eigenvectors of the linear renormalization group transformation for observables along the renormalized trajectory of the hierarchical non-linear $O(N)$-invariant $\sigma$-model by means of perturbation theory in the running coupling constant. Moving eigenvectors are defined as solutions to a Callan-Symanzik type equation.