This paper develops a new theory of tensor invariants of a completely integrable non-degenerate Hamiltonian system on a smooth manifold M(n). The central objects in this theory are supplementary invariant Poisson structures P-c which are incompatible with the original Poisson structure P-1 for this Hamiltonian system. A complete classification of invariant Poisson structures is derived in a neighbourhood of an invariant toroidal domain. This classification resolves the well-known Inverse Problem that was brought into prominence by Magri's 1978 paper devoted to the theory of compatible Poisson structures. Applications connected with the KAM theory, with the Kepler problem, with the basic integrable problem of celestial mechanics, and with the harmonic oscillator are pointed out. A cohomology is defined for dynamical systems on smooth manifolds. The physically motivated concepts of dynamical compatibility and strong dynamical compatibility of pairs of Poisson structures are introduced to study the diversity of pairs of Poisson structures incompatible in Magri's sense. It is proved that if a dynamical system V preserves two strongly dynamically compatible Poisson structures P-1 and P-2 in a general position then this system is completely integrable. Such a system V generates a hierarchy of integrable dynamical systems which in general are not Hamiltonian neither with respect to P-1 nor with respect to P-2. Necessary conditions for dynamical compatibility and for strong dynamical compatibility are derived which connect these global properties with new local invariants of an arbitrary pair of incompatible Poisson structures.
