In the extremal case when $k=n-1$, constructing Lagrangian fibrations using Proposition 5.2 is very easy. In this situation, the reduced spaces $X_{t}$ are two dimensional and every map $G_{t}:X_{t}\rightarrow\mathbb{R}$ with $1$-dimensional level sets defines a Lagrangian fibration on $X_{t}$. In particular, any $T^{n-1}$-invariant continuous map $G:X\rightarrow\mathbb{R}$ which, on each $X_{t}$, descends to a map $G_{t}$ with $1$-dimensional level sets can be used to construct Lagrangian fibrations. We will make much use of this fact later on.