The generalized Calabi ansatz has two familiar special cases:

For $m=n$, then $Y$ is just a point, and $0<r_{i}\ll 1$ amounts to the subset $|\xi_{i}|\ll 1$ inside $(\mathbb{C}^{*})^{n}$. The NA MA equation is just the real MA equation

$$\det(D^{2}u)=\text{const}.$$

The ansatz then produces a Calabi-Yau metric, known as the semiflat metric, familiar in the SYZ conjecture [15].

For $m=1$, and $L$ trivial, then $Z$ is the total space of a positive line bundle $L_{1}$ over an $(n-1)$-dimensional Calabi-Yau manifold $Y$. We then take $h_{L_{1}}$ to be the Hermitian metric corresponding to the Calabi-Yau metric in $c_{1}(L_{1})$. We can also view $h_{L_{1}}$ as a function on $Z$, equal to $r_{L_{1}}^{2}$. Up to a normalising constant, the Calabi ansatz is

$$\omega_{Cal}=\frac{n}{n+1}dd^{c}(-\log h_{L_{1}}^{1/2})^{(n+1)/n}.$$

We compare this to the NA MA equation (4) in this case: $u$ is a function of a single real variable $x$, satisfying

$$u^{\prime\prime}u^{\prime n-1}=\text{const}.$$

Up to constant $u=x^{(n+1)/n}$. Thus the NA MA equation reproduces the Calabi ansatz.