  Motivated by the Lagrange top coupled to an oscillator, we consider the
quasi-periodic Hamiltonian Hopf bifurcation. To this end, we develop the normal
linear stability theory of an invariant torus with a generic (i.e.,
non-semisimple) normal $1:-1$ resonance. This theory guarantees the persistence
of the invariant torus in the Diophantine case and makes possible a further
quasi-periodic normal form, necessary for investigation of the non-linear
dynamics. As a consequence, we find Cantor families of invariant isotropic tori
of all dimensions suggested by the integrable approximation.
