  In this paper, we consider a classical Hamiltonian normal form with
degeneracy in normal direction. In previous results, one needs to assume that
the perturbation satisfies certain non-degenerate conditions in order to remove
the degeneracy in the normal form. In stead of that, we introduce a topological
degree condition and a weak convexity condition, which are easy to be verified,
and we prove the persistence of lower dimensional tori without any restriction
on perturbation but only smallness and analyticity.
