Let $X^{\prime}=\mathbb{C}^{2}-\{z_{1}z_{2}-1=0\}$ and let $X=X^{\prime}\times\mathbb{C}^{\ast}$ with the standard symplectic structure. Define $f:X\rightarrow\mathbb{R}^{3}$ by $f=(f_{1},f_{2},f_{3})$ where

$f_{1}=\frac{|z_{1}|^{2}-|z_{2}|^{2}}{2}$,

$f_{2}=\log|z_{3}|$,

$f_{3}=\log|z_{1}z_{2}-1|$.

Again, these functions Poisson commute, hence $f$ is Lagrangian. The singular fibres of $f$ are lying over $\Delta=\{(0,r,0)\mid r\in\mathbb{R}\}$. The reader may verify that the above gives a generic-singular fibration.