I want to know if there exists examples of Calabi-Yau manifolds with $\mathbb{T}^2$-invariant $SU(n)$-structure. In particular these actions are both Killing and holomorphic. I am especially interested in the case when $n\geq4$. I would be happy even with local examples.

A CY $n$-fold (for me) is a Kahler manifold with holonomy group equal to $SU(n)$, in particular I don't include products of lower dimensional CY manifolds. Since CY manifolds are Ricci flat, this implies that these examples (if any) cannot be compact unless they are the products of a CY $(n-1)$-fold and $\mathbb{T}^2$ (elliptic curve). So the question is asking for non-compact examples.

In the case of $S^1$-invariant CY manifolds, I am aware of a few examples; in real dimension $4$ the Eguchi-Hanson and Taub-Nut metrics, and in higher dimensions there are examples on line bundles over CY manifolds and higher dimensional version of "Taub-Nut metrics". I would also be interested in knowing more examples of those as well.