Invariants of symbols of the linear differential operators

In this paper we classify the symbols of the linear differential operators of order $k$, which act from the module $C^\infty(\xi)$ to the module $C^\infty(\xi^t)$, where $\xi\colon E(\xi)\to M$ is vector bundle over the smooth manifold $M$, bundle $\xi^t$ is either $\xi^*$ with fiber $E^*:=\mathrm{Hom}(E,\mathbb{C})$ or $\xi^\flat$ with fiber $E^\flat:=\mathrm{Hom}(E, \Lambda^n T^*)$ and $C^\infty(\xi)$, $C^\infty(\xi^t)$ are the modules of their smooth sections. To find invariants of the symbols we associate with every non-degenerated symbol the tuple of linear operators acting on space $E$ and reduce our problem to the classification of such tuples with respect to some orthogonal transformations. Using the results of C. Procesi, we find generators for the field of rational invariants of the symbols and in terms of these invariants provide a criterion of equivalence of non-degenerated symbols.