Canonical forms for the invariant tensors and $A$-$B$-$C$-cohomologies of integrable Hamiltonian systems

The canonical forms for the $(\ell ,m)$ tensors, $\ell +m\leqslant 3$, that are invariant with respect to a Liouville-integrable non-degenerate Hamiltonian system $V$ on a symplectic manifold $M^{2k}$ are derived. It is proved that the characteristic polynomial of any invariant $(1,1)$ tensor $A^\alpha _\beta$ is a perfect square; therefore its eigenvalues have even multiplicities. Any invariant metric $g_{\alpha \beta }$ is indefinite and has signature $\sigma \leqslant k$. The derived canonical forms are applied to the calculation of the $A$-$B$-$C$-cohomologies of Liouville-integrable Hamiltonian systems.