 Solution

For any positive-dimensional $X$, choose a smooth function $h$ with $dh$ not identically zero. A nonconstant smooth bump in a coordinate chart supplies one on an arbitrary smooth manifold. Translate by this exact differential form:The cotangent fiber translation identities giveThus a nontrivial exact fiber translation is a symplectomorphism that fails to preserve the Liouville one-form. For example, on $T^{\ast }\mathbb{R}$ one can take $T(q,p)=(q,p+1)$: it preserves $dq\wedge dp$ but changes $pdq$ to $(p+1)dq$.

If dimension zero is admitted, $T^{\ast }X=X$ and $\alpha =0$, so every diffeomorphism preserves $\alpha$ and this requested distinction is impossible. The example therefore applies in positive dimension, as required for a nonzero differential one-form.