  We consider a dimensional reduction of the (deformed) Hermitian Yang-Mills
condition on $S^1$-invariant K\"ahler Einstein $6$-manifolds. This allows us to
reformulate the (deformed) Hermitian Yang-Mills equations in terms of data on
the quotient K\"ahler $4$-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 $\mathcal{O}_{\mathbb{C}\mathbb{P}^2}(-3)$.
