A simple example of stitched Lagrangian fibration which can be constructed using Theorem 6.19 is as follows. Define the sequence $\ell^{c}$ to be identically zero and choose the terms of $\ell^{d}$ and $\ell^{e}$ to be zero except the first order ones, which we define to be

$$\ell_{1}^{d}=m_{1}\,dy_{2}\ \ \text{and}\ \ \ell_{1}^{e}=m_{2}\,dy_{3}.$$

Clearly $\ell_{1}^{c}$, $\ell_{1}^{d}$ and $\ell_{1}^{e}$ satisfy the integral conditions of Theorem 6.19, moreover they are fibrewise constant, therefore they define fake stitched fibrations. Since the fibration is smooth after a change of coordinates on the base, it induces an affine structure on the base. One can easily see that in the case $m_{1}=-1$ and $m_{2}=1$ and $U=\mathbb{R}^{3}-\Delta$, this affine structure is simple and affine isomorphic to a negative vertex of Example 3.12. Notice that we could also replace $\Delta$ with $\Delta_{\tau}$ and obtain an affine structure which is isomorphic to the one in Example 3.13.