their stability and the procedure of solving the Hermitian-Yang-Mills equations were discussed e.g. in [@BB; @Yau; @Yaugroup; @Fern]. Here we consider the procedure of solving the Hermitian-Yang-Mills equations on the homogeneous nearly Kähler spaces Sp(2)/Sp(1)$\times$U(1), SU(3)/U(1)$\times$U(1) and nearly Calabi-Yau spaces Sp(1,1)/Sp(1)$\times$U(1), SU(1,2)/U(1)$\times$U(1) which may serve as a local model for compact case obtained by quotienting out by a discrete group.

The above-mentioned four manifolds are twistor spaces of the four-dimensional sphere $S^4$, projective plane ${\mathbb C}P^2$ and balls $B_4=\ $Sp(1,1)/Sp(1)$\times$Sp(1), ${\mathbb C}B_2=\ $SU(1,2)/S(U(1)$\times$U(2)) endowed with nonintegrable almost complex structure. Hence, complex vector bundles over these twistor spaces can carry a pseudo-holomorphic structure but not the holomorphic ones. That is why we begin our discussion with the notion of the pseudo-holomorphic bundles [@Bryant] and their relations with the Hermitian-Yang-Mills equations. Then we consider the twistor space ${{\cal T}}(M^4)$ of an oriented Riemannian 4-manifold $M^4$ along with the canonical projection $\pi : {{\cal T}}(M^4)\to M^4$ and a complex vector bundle $E$ with an anti-self-dual connection $A$. We show that any anti-self-dual gauge field $F=\diff A+A\wedge A$ on $M^4$ uplifted to the gauge field $\hat F:=\pi^*F$ on ${{\cal T}}(M^4)$ provides a solution to the Hermitian-Yang-Mills equations on ${{\cal T}}(M^4)$. This correspondence allows us 