Explicit abelian instantons on $S^1$-invariant Kähler Einstein $6$-manifolds

We consider a dimensional reduction of the (deformed) Hermitian Yang-Mills condition on $S^1$-invariant Kähler Einstein $6$-manifolds. This allows us to reformulate the (deformed) Hermitian Yang-Mills equations in terms of data on the quotient Kähler $4$-manifold. In particular, we apply this construction to the canonical bundle of $\mathbb{C}{\mathbb{P}}^2$ endowed with the Calabi ansatz metric to find explicit abelian $SU(3)$ instantons and we show that these are determined by the spectrum of $\mathbb{C}{\mathbb{P}}^2$. We also find $1$-parameter families of explicit deformed Hermitian Yang-Mills connections. As a by-product of our investigation we find a coordinate expression for its holomorphic volume form which leads us to construct a special Lagrangian foliation of ${\mathcal{O}}_{\mathbb{C}{\mathbb{P}}^2}(-3)$.