Consider the multidimensional annulus $\{(p,\theta)\} = \mathbb R^n\times\mathbb T^n$ endowed by the $1$-form $\omega=p\,d\theta$. A diffeomorphism $A$ of this annulus onlo itself is said to be exact symplectic if the $1$-form $A^{\ast}\omega-\omega$ is exact, and the $n$-torus of the form $p=f(\theta)$ is said to be Lagrangian if the $1$-form $f(\theta)\,d\theta$ is closed.
The following very easy fact is well known: if a diffeomorphism A is exact symplectic and close to the multidimensional rotation by a varying angle: $(p,\theta)\mapsto(p,\theta+\partial S(p)/\partial p)$, then any Lagrangian torus $p=f(\theta)$ close to $p=\mathrm{const}$ intersects its $A$-image (in fact, more general statements hold).
For each $n\geqslant 2$ it is also very easy to construct an exact symplectic diffeomorphism $A:(p,\theta)\mapsto(p,\theta+\mathrm{const})$ (the multidimensional rotation by a constant angle) and a non-Lagrangian torus $p=f(\theta)$ that is arbitrarily close to $p=\mathrm{const}$ and does not intersect its $A$-image.
The question: are such examples known in the literature? If yes, I'd like to have a relevant reference.
