  In this paper, we investigate Kolmogorov type theorems for small
perturbations of degenerate Hamiltonian systems. These systems are index by a
parameter $\xi$ as \( H(y,x,\xi) = \langle\omega(\xi),y\rangle + \varepsilon
P(y,x,\xi,\varepsilon) \) where $\varepsilon>0$. We assume that the frequency
map, $\omega$, is continuous with respect to $\xi$. Additionally, the
perturbation function, $P(y,x,\cdot, \varepsilon)$, maintains H\"{o}lder
continuity about $\xi$. We prove that persistent invariant tori retain the same
frequency as those of the unperturbed tori, under certain topological degree
conditions and a weak convexity condition for the frequency mapping. Notably,
this paper presents, to our understanding, pioneering results on the KAM
theorem under such conditions-with only assumption of continuous dependence of
frequency mapping $\omega$ on the parameter.
