Consider a smooth proper Lagrangian fibration $f:X\rightarrow B$, with $B=\mathbb{R}\times M$ and $f=(\mu,G)$, where $\mu$ is the moment map of a free $S^{1}$ action and $G$ is $S^{1}$ invariant. Assuming $B$ is contractible and having chosen bases $\gamma^{\pm}$ of $H_{1}(X,\mathbb{Z})$ as in $(a)$ and $(b)$ above, on $B$ we can apply the admissible change of coordinates $\alpha$ as in Proposition 6.5. Clearly $f^{\prime}=\alpha\circ f$ is (tautologically) a fake stitched fibration. Given a Lagrangian section $\sigma$ of $f^{\prime}$, it easy to see that the normal form for $(X,f^{\prime}(B),f^{\prime},\gamma^{+},\sigma)$ is of the type $(Y,U,f_{u},\gamma_{0},\sigma_{0})$ where $Y=T^{*}U/\Lambda$ and

$$u(y_{1},\ldots,y_{n},b_{1},\ldots,b_{n})=(b_{1},b_{2}-m_{2}b_{1},\ldots,b_{n}-m_{n}b_{1}),$$

i.e. the projection composed with a linear change of coordinates. In this case the only non-zero invariant is $\ell_{1}$ which is given by

$$\ell_{1}=\sum_{j}\,m_{j}dy_{j}.$$

Clearly $\ell_{1}$ is fibrewise constant.