Let $\Phi$, $X_{t}$ and $G_{t}$ be as defined above. Let $Q$ be the map given by

$$Q(t,u_{1},\ldots,u_{n-1})=(t,G_{t}(u_{1},\ldots,u_{n-1})).$$

(32)

Then $Q$ is defined on the dense open subset $Y\subseteq\mathbb{R}\times\mathbb{C}^{n-1}$ defined by

$$Y=\{(t,u_{1},\ldots,u_{n-1})\in\mathbb{R}\times\mathbb{C}^{n-1}\ |\ (u_{1},\ldots,u_{n-1})\in X_{t}\}.$$

Letting $\bar{\pi}$ be as in (26) and

$$X=(\bar{\pi})^{-1}(Y)$$

with the standard symplectic form induced from $\mathbb{C}^{n}$, the map $f:X\rightarrow\mathbb{R}^{n}$ given by

$$f=Q\circ\bar{\pi}$$

is a piecewise smooth Lagrangian fibration of $X$ which fails to be smooth on the $(2n-1)$-dimensional subspace $\mu^{-1}(0)\cap X$.