Since $\alpha_{1}(b)=b_{1}$, it is enough to show that, if for fixed $t\in\mathbb{R}$ we let $U_{t}=\{b_{1}=t\}$, then $\alpha_{t}=\alpha|_{U_{t}}$ is a bijection onto its image. If $\lambda_{2}$ and $\lambda_{3}$ are the periods of the fibration corresponding to $\gamma_{2}$ and $\gamma_{3}$, then $\alpha_{t}$ is computed by taking primitives of $\lambda_{2}|_{U_{t}}$ and $\lambda_{3}|_{U_{t}}$. If we let $X_{t}$ denote the symplectic reduction of $X$ at $t$ and $G_{t}:X_{t}\rightarrow\mathbb{R}^{2}$ the reduced fibration, then it is not difficult to see that $\lambda_{2}|_{U_{t}}$ and $\lambda_{3}|_{U_{t}}$ are in fact periods of $G_{t}$ (cf. [3]Lemma 5.9). Now the conclusion follows by simply observing that $G_{t}$ is a proper Lagrangian submersion, i.e. an integrable system. The argument works also when $t=0$.

An explicit computation of the periods was done in [3]Proposition 5.10 for the fibration in Example 5.8. There we found that

where $\beta_{1}$ and $\beta_{2}$ are functions depending only on $b_{1}$. The periods of the perturbed fibration obtained in Lemma 7.6 will have this same expression away from where the perturbation took place (i.e. away from the white region in Figure 15), for example in a neighborhood of the codimension 1 part of $\Delta$. It is easy to see from this expression of the periods that $\alpha$ extends continuously to $\Delta$ and that it is a bijection. ∎