  We study the problem of conservation of maximal and lower-dimensional
invariant tori for analytic convex quasi-integrable Hamiltonian systems. In the
absence of perturbation the lower-dimensional tori are degenerate, in the sense
that the normal frequencies vanish, so that the tori are neither elliptic nor
hyperbolic. We show that if the perturbation parameter is small enough, for a
large measure subset of any resonant submanifold of the action variable space,
under some generic non-degeneracy conditions on the perturbation function,
there are lower-dimensional tori which are conserved. They are characterised by
rotation vectors satisfying some generalised Bryuno conditions involving also
the normal frequencies. We also show that, again under some generic assumptions
on the perturbation, any torus with fixed rotation vector satisfying the Bryuno
condition is conserved for most values of the perturbation parameter in an
interval small enough around the origin. According to the sign of the normal
frequencies and of the perturbation parameter the torus becomes either
hyperbolic or elliptic or of mixed type.
