Consider the symplectomorphism $\Phi$ from Example 5.5. The reduced fibration at time $t=0$, i.e. the map $G_{0}=\Log\circ\Phi\circ\Gamma_{0}$, has many Lagrangian sections, since the $\Log$ fibration has many. In particular we can choose one which does not intersect $\Sigma=\Crit(f)$, this follows for example by observing that the following Lagrangian section of the $\Log$ fibration

does not intersect the surface $\Sigma^{\prime}=\{v_{1}+v_{2}+1\}$. It is easy to see that a section which does not intersect $\Sigma$ can be lifted to $\mu^{-1}(0)$. The image of this lift is a coisotropic $2$ dimensional submanifold of $X$. Applying the coisotropic embedding theorem, we can extend this submanifold to a Lagrangian submanifold along a direction which is transversal to $\mu^{-1}(0)$, e.g. along $i\eta$, where $\eta$ is the Hamiltonian vector field of the $S^{1}$ action. This submanifold is then the image of a section of the fibration in Example 5.5.

In the case of $\Phi$ from Example 5.8, $\Phi(\Sigma)$ is a small perturbation of $\Sigma^{\prime}$ as above. One can see that the section in (47) also avoids $\Phi(\Sigma)$. Then the argument follows as before. ∎