Nearly integrable Hamiltonian systems are considered, for which the unperturbed system has a lower dimensional torus not satisfying the second Mel'nikov condition; on a 2 : 1 covering space a suitable choice of the toral angles yields a vanishing Floquet exponent. A nilpotent Floquet matrix leads to the quasi-periodic analogue of the period-doubling bifurcation, so particular emphasis is given to the case of vanishing normal linear behaviour. The actions conjugate to the toral angles unfold the various ways in which the degenerate torus becomes normally elliptic, hyperbolic or parabolic. With a KAM-theoretic approach it is then shown that this bifurcation scenario survives a non-integrable perturbation, parametrized by pertinent large Cantor sets. The bifurcation scenario is governed by , the ‘first’ unimodal planar singularity, which has co-dimension 7 with respect to all planar singularities. In the present context this high number is reduced to co-dimension 3 since the π-rotation on the 2 : 1 covering space has to be respected, and in case the Hamiltonian system is reversible there is a further reduction by 1 and the co-dimension becomes 2. In such low co-dimensions it becomes more transparent why the modulus μ – although playing a prominent role during the KAM iteration – is of limited influence on the dynamical implications.