is said to define a pseudo-holomorphic structure if it has curvature ${{\cal F}}=
\diff{{\cal A}}+{{\cal A}}\wedge{{\cal A}}$ of type (1,1) with respect to (w.r.t.) ${{\cal J}}$, i.e. $$\label{2.4}
{{\cal F}}^{0,2}=0={{\cal F}}^{2,0}\ .$$

One can endow the bundle ${{\cal E}}$ with a Hermitian metric and choose ${{\cal A}}$ to be compatible with the Hermitian structure on ${{\cal E}}$. If, in addition, $\o$ is an almost Hermitian structure on $(X^{2n}, {{\cal J}})$ and $$\label{2.5}
\o\,\lrc\,{{\cal F}}= \im\, \la\, \rm{Id}_{{{\cal E}}}$$ with $\la\in{\mathbb R}$, the connection ${{\cal A}}$ is said to be (pseudo-)Hermitian-Yang-Mills [@Bryant]. We shall consider (\[2.5\]) with $\la =0$, i.e. assume $c_1({{\cal E}})=0$ since for a bundle with field strength ${{\cal F}}$ of non-zero degree one can obtain a zero-degree bundle by considering $\widetilde{{\cal F}}={{\cal F}}- \sfrac{1}{k}(\tr{{\cal F}})\cdot
{\bf 1}_k$, where $k=\,$rank$\,{{\cal E}}$.

[**Hermitian-Yang-Mills equations**]{}. The Hermitian-Yang-Mills (HYM) equations[^1] read $$\label{2.6}
{{\cal F}}^{0,2}=0\und\o\,\lrc\,{{\cal F}}=0\ .$$ In the special case of an almost complex 4-manifold $X^4$ with a metric $g$ they coincide with the anti-self-dual Yang-Mills (ASDYM) equations $$\label{2.7}
*{{\cal F}}=-{{\cal F}}\ ,$$ where $*$ is the Hodge operator. Note that (\[2.7\]) is valid on manifolds $(M^4, g)$ which are not necessarily almost complex manifolds.

Recall that there are various generalizations of 