Consider the following affine symplectomorphism of $(\mathbb{C}^{2},\omega_{\mathbb{C}^{2}})$

$$\Phi:(u_{1},u_{2})\mapsto(-u_{2},u_{1}-1).$$

(37)

The surface $\Sigma$ is sent by $\Phi\circ\Gamma_{0}$ to $\Sigma^{\prime}=\{v_{2}+1=0\}$. The amoeba of $\Sigma^{\prime}$ is just a straight line. The resulting fibration $f$ is

$$f(z_{1},z_{2},z_{3})=\left(\frac{|z_{1}|^{2}-|z_{2}|^{2}}{2},\,\log|z_{3}|,\,\log|\gamma(z_{1},z_{2})-1|\right).$$

(38)

The discriminant locus is $\{0\}\times\mathbb{R}\times\{0\}\subset\mathbb{R}^{3}$, a horizontal line in the plane $\{0\}\times\mathbb{R}^{2}$. The fibration is a piecewise smooth version of the generic singular fibration in Example 4.5. Notice that this fibration is invariant under the Hamiltonian $T^{2}$-action

$$(e^{i\theta_{1}},e^{i\theta_{2}})\cdot(z_{1},z_{2},z_{3})=(e^{i\theta_{1}}z_{1},\,e^{-i\theta_{1}}z_{2},\,e^{2i\theta_{2}}z_{3}),$$

(39)

whose moment map is

$$(z_{1},z_{2},z_{3})\mapsto\left(\frac{|z_{1}|^{2}-|z_{2}|^{2}}{2},\,|z_{3}|^{2}\right).$$

There are other choices of symplectomorphisms $\Phi$ giving piecewise smooth generic fibrations. Although not very different from the previous one, we will write other two for convenience, since we will need them in the next example. The first one is

$$\Phi:(u_{1},u_{2})\mapsto(u_{1}-1,u_{2}-\sqrt{2}).$$

(40)

It gives the fibration

$$f(z_{1},z_{2},z_{3})=\left(\frac{|z_{1}|^{2}-|z_{2}|^{2}}{2},\,\log|\gamma(z_{1},z_{2})-1|,\,\log\left|z_{3}-\sqrt{2}\right|\right),$$

(41)

whose discriminant locus is the vertical line $\{0\}\times\{0\}\times\mathbb{R}\subset\mathbb{R}^{3}$. Also in this case it is clearly invariant under a $T^{2}$ action. The last choice of $\Phi$ is

$$\Phi:(u_{1},u_{2})\mapsto\frac{1}{\sqrt{2}}(u_{1}-u_{2},\,u_{1}+u_{2}),$$

(42)

giving

$$f(z_{1},z_{2},z_{3})=\left(\frac{|z_{1}|^{2}-|z_{2}|^{2}}{2},\,\log|\gamma(z_{1},z_{2})-z_{3}|,\,\log|\gamma(z_{1},z_{2})+z_{3}|\right),$$

(43)

whose discriminant is the slope +1 diagonal through zero in $\{0\}\times\mathbb{R}^{2}$. The $T^{2}$ action in this case is given by

$$(e^{i\theta_{1}},e^{i\theta_{2}})\cdot(z_{1},z_{2},z_{3})=(e^{i(\theta_{2}+\theta_{1})}z_{1},\,e^{i(\theta_{2}-\theta_{1})}z_{2},\,e^{2i\theta_{2}}z_{3}),$$

(44)

whose moment map is

$$(z_{1},z_{2},z_{3})\mapsto\left(\frac{|z_{1}|^{2}-|z_{2}|^{2}}{2},\,\frac{|z_{1}|^{2}+|z_{2}|^{2}}{2}+|z_{3}|^{2}\right).$$

In the above examples, the reduced spaces are all 2-dimensional. Using Remark 5.3 we can construct variations of (38) by replacing the last component of (38) with any function depending on $t=\frac{|z_{1}|^{2}-|z_{2}|^{2}}{2}$, $s=|z_{3}|^{2}$ and $u_{1}=z_{1}z_{2}$, subject to the condition that all the maps $G_{t}$ have $1$-dimensional level sets. A choice providing an example of a smooth fibration is given by $G=\log|u_{1}-1|$, which gives us Example 4.5. One can do more. In fact, one can take a function $G$ which gives an interpolation between the piecewise smooth fibration in (38) and the smooth one in Example 4.5. This can be done by taking $G$ depending also on $s$, such that $G$ is equal to $\log|\gamma(z_{1},z_{2})-1|$ when $s$ is big and equal to $\log|u_{1}-1|$ when $s$ is small. We will say more about this later on, as this idea is useful in an important step of the main construction of the paper.