An example of a stitched Lagrangian fibration constructed using Theorem 6.15 is the following. We can choose the elements of the sequence $\ell^{u}$ to be all zero, while the elements of the sequence $\ell^{d}$ to be all zero except $\ell_{1}^{d}$ which we define to be

$$\ell_{1}^{d}=m\,dy_{2}.$$

It is clear that the resulting fibration is only fake stitched, in fact the invariants are fibrewise constant. One can also see that, in the case $m=1$ and $U=\mathbb{R}^{2}-\{0\}$, the fibration is symplectically conjugate to $(X,\alpha\circ f)$, where $(X,f)$ is a smooth focus-focus fibration (where the singular fibre has been removed) and $\alpha$ is the action coordinates map (see the discussion after Example 3.20 and Example 6.13). In particular this fibration induces an affine structure on the base which is simple.