The symplectic shift of the cylinder changes $\lambda$ by the closed, non-exact form $ad\theta$ and has no global generating function

Clause 1: the lift $(\theta ,I)\mapsto (\theta ,I+a)$ and its inverse $(\theta ,I)\mapsto (\theta ,I-a)$ descend and are smooth in the product lift charts, so $\mathrm{\Phi }$ is a smooth diffeomorphism. By [F2] and [F1], ${\pi }^{\ast }({\mathrm{\Phi }}^{\ast }{\omega }_C)=(\mathrm{\Phi }\circ \pi {)}^{\ast }{\omega }_C=(\pi \circ \tilde{\mathrm{\Phi }}{)}^{\ast }{\omega }_C={\tilde{\mathrm{\Phi }}}^{\ast }({\pi }^{\ast }{\omega }_C)={\tilde{\mathrm{\Phi }}}^{\ast }(d\theta \wedge dI)=d\theta \wedge dI={\pi }^{\ast }{\omega }_C$, so injectivity of ${\pi }^{\ast }$ gives ${\mathrm{\Phi }}^{\ast }{\omega }_C={\omega }_C$; likewise ${\pi }^{\ast }({\mathrm{\Phi }}^{\ast }{\lambda }_C)={\tilde{\mathrm{\Phi }}}^{\ast }(Id\theta )=(I+a)d\theta =Id\theta +ad\theta ={\pi }^{\ast }({\lambda }_C+ad{\theta }_C)$, which gives ${\mathrm{\Phi }}^{\ast }{\lambda }_C={\lambda }_C+ad{\theta }_C$.

Clause 2, closedness: $d(ad{\theta }_C)=ad(d{\theta }_C)=0$ because ${\pi }^{\ast }(d(d{\theta }_C))=d(d\theta )=0$ and ${\pi }^{\ast }$ is injective; so $ad{\theta }_C$ is closed.

Clause 2, non-exactness: suppose $W:C\to \mathbb{R}$ is $C^1$ with $dW=ad{\theta }_C$, and set $g(t):=W(\pi (t,0))$ for $t\in [0,1]$. By the chain rule and $d{\theta }_C({\mathrm{\partial }}_{\theta })=1$ at the image point, $g$ is $C^1$ with $g^{\mathrm{\prime }}(t)=a$ for $0<t<1$; by [F3], $g(1)-g(0)=a$. But $\pi (1,0)=\pi (0,0)$ in $C$, so $g(1)=g(0)$; this contradicts $a\mathrm{\ne }0$. Hence no such $W$ exists, and ${\mathrm{\Phi }}^{\ast }{\lambda }_C-{\lambda }_C=ad{\theta }_C$ is a closed form representing a nonzero class.

Clause 3, global failure: suppose a $C^1$ function $S$ exists on $(\mathbb{R}\mathrm{/}\mathbb{Z})\times \mathbb{R}$ whose generated second-kind map is $\mathrm{\Phi }$, that is, with $p=S_{\theta }=I=P-a$ and $Q=S_P$ equal to the circle coordinate $\theta$ at every point, the former because $\mathrm{\Phi }$ carries $(\theta ,I)$ to $(\theta ,I+a)$, so the new momentum is $P=I+a$ and the old momentum is $I=P-a$. Lift $S$ to a $1$-periodic $C^1$ function $\widehat{S}$ on ${\mathbb{R}}^2$. The relation $Q=\theta$ reads ${\widehat{S}}_P(\theta ,P)-\theta \in \mathbb{Z}$; since this difference is continuous and integer-valued on ${\mathbb{R}}^2$, it is constant: if it took two distinct values, the intermediate value theorem applied along the segment joining the corresponding points would produce a non-integer value. Write the constant as $m\in \mathbb{Z}$, so ${\widehat{S}}_P(\theta ,P)=\theta +m$; the fundamental theorem of calculus in $P$ gives $\widehat{S}(\theta ,P)=\theta P+mP+c(\theta )$ with $c(\theta ):=\widehat{S}(\theta ,0)$. Differentiating in $\theta$ gives ${\widehat{S}}_{\theta }(\theta ,P)=P+c^{\mathrm{\prime }}(\theta )=P-a$, so $c^{\mathrm{\prime }}(\theta )=-a$ and, by [F3] in the variable $\theta$, $c(1)-c(0)=-a\mathrm{\ne }0$, contradicting the $1$-periodicity $c(1)=c(0)$ of the lift. Hence no global $S$ exists, while [F4] exhibits the local generating function $S(\theta ,P)=\theta P-a\theta$ on every chart where $\theta$ is a single-valued coordinate.

First-kind exclusion: a first-kind generating function cannot generate $\mathrm{\Phi }$ even locally. In fixed input and output lift charts, the unchanged circle coordinate means $Q=\theta +k$ for a locally constant integer $k$; after restricting to a connected neighbourhood it is fixed. On an open input phase neighbourhood, fix $\theta$ and choose two distinct old momenta $I_1,I_2$ in its vertical interval. The first-kind relation $I=S_{\theta }(\theta ,Q)$ would give both $I_1$ and $I_2$ the same value $S_{\theta }(\theta ,\theta +k)$, a contradiction. Therefore no global first-kind generator can restrict to local generating data for this shift. Together with step 3.1, this excludes global generators of either kind defined in [F4].