There are a number of recent constructions of complete Calabi-Yau metrics on $\mathbb{C}^{n}$ with $n\geq 3$, that share the unifying theme of holomorphic fibrations. The works [12][20][6] start with a holomorphic fibration given by a weighted homogeneous polynomial $F:\mathbb{C}^{n}\to\mathbb{C}$, such that $F^{-1}(0)$ carries a Sasakian-Einstein cone metric. The other fibres carry asymptotically conical Calabi-Yau metrics modelled at infinity on $F^{-1}(0)$. From this, one builds a Calabi-Yau metric on $\mathbb{C}^{n}$, whose fibrewise restrictions are approximated by these Calabi-Yau metrics on the fibres outside of a compact region, and in the horizontal direction is approximated by the pullback of the Euclidean metric on $\mathbb{C}$. Such metrics on $\mathbb{C}^{n}$ have maximal volume growth, and the tangent cone at infinity is $F^{-1}(0)\times\mathbb{C}$ with the product metric. From an algebro-geometric perspective, the singularities on $F^{-1}(0)$ are klt, which should be viewed as mild singularities. The coordinate functions on $\mathbb{C}^{n}$ all have polynomial growth with respect to the geodesic distance to the origin, even though the growth rates are typically not linear.

More recently a family of new Taub-NUT type Calabi-Yau metrics were constructed on $\mathbb{C}^{3}$ [13], using a generalized Gibbons-Hawking framework. The asymptotic geometry near infinity is generically a $T^{2}$-fibration over $\mathbb{R}^{4}$, but along three rays inside $\mathbb{R}^{4}$ the metric looks like a Taub-NUT fibration over a cylinder. These metrics are fundamentally different in that it has volume growth order $Vol(B(r))\sim O(r^{4})$ coming from the $\mathbb{R}^{4}$ direction, which is not maximal volume growth. Algebro-geometrically, these metrics are associated with the holomorphic fibration

$$F(z_{1},z_{2},z_{3})=z_{1}z_{2}z_{3}:\mathbb{C}^{3}\to\mathbb{C},$$

whose fibres are generically cylinders, contributing two dimensions to $T^{2}$ and two other dimensions to $\mathbb{R}^{4}$. The $\mathbb{C}$ factor contributes the other two dimensions to $\mathbb{R}^{4}$. Notice the singular fibres are reducible, and the nature of the singularity is much worse than klt. In terms of the growth of holomorphic functions, only $z_{1}z_{2}z_{3}$ has polynomial growth, while $z_{1},z_{2},z_{3}$ individually all have exponential type growth, meaning that $\log|z_{k}|$ has polynomial growth.

In view of these constructions, the new feature of this paper is that in the generalized Calabi ansatz, the approximately Calabi-Yau fibres do not appear as fibres of holomorphic fibrations, but rather come from the effective description of an iterated non-holomorphic fibration. The algebraic functions on $X$ have exponential type growth. The prototype of these phenomena is of course already known in the case of the Calabi ansatz, but we believe the NA MA equation points towards a much larger generality of examples, not limited to ODE reduction methods.