  Given any symplectomorphism on $D^{2n} (n\geq 1)$ which is $C^{\infty}$ close
to the identity, and any completely integrable Hamiltonian system $\Phi^t_H$ in
the proper dimension, we construct a $C^{\infty}$ perturbation of $H$ such that
the resulting Hamiltonian flow contains a "local Poincar\'{e} section" that
"realizes" the symplectomorphism. As a (motivating) application, we show that
there are arbitrarily small perturbations of any completely integrable
Hamiltonian system which are entropy non-expansive (and, in particular, exhibit
hyperbolic behavior on a set of positive measure).
