  We prove the existence of invariant tori to the area-preserving maps defined
on $ \mathbb{R}^2\times\mathbb{T} $
  \begin{equation*}
  \bar{x}=F(x,\theta),
  \qquad \bar{\theta}=\theta+\alpha\, \,(\alpha\in
\mathbb{R}\setminus\mathbb{Q}),
  \end{equation*} where $ F $ is closed to a linear rotation, and the
perturbation is ultra-differentiable in $ \theta\in \mathbb{T},$ which is very
closed to $C^{\infty}$ regularity. Moreover, we assume that the frequency
$\alpha$ is any irrational number without other arithmetic conditions and the
smallness of the perturbation does not depend on $\alpha$. Thus, both the
difficulties from the ultra-differentiability of the perturbation and
Liouvillean frequency will appear in this work. The proof of the main result is
based on the Kolmogorov-Arnold-Moser (KAM) scheme about the area-preserving
maps with some new techniques.
