while a manifold $M^4$ admits in general no almost complex structure, its twistor space ${{\cal T}}(M^4)$ can always be equipped with two natural almost complex structures. The first, ${{\cal J}}={{\cal J}}_+$, introduced in [@AHS], is integrable if and only if the Weyl tensor of $g$ on $M^4$ is anti-self-dual, while the second ${{\cal J}}={{\cal J}}_-$, introduced in [@ES], is never integrable. In fact, ${{\cal J}}_+$ and ${{\cal J}}_-$ differ only on $S^2_x\cong{\mathbb C}P^1\hra{{\cal T}}(M^4)$ (${{\cal J}}_+|_{{\mathbb C}P^1}
= -{{\cal J}}_-|_{{\mathbb C}P^1}$) and coincide on $T_xM^4$. Twistor spaces ${{\cal T}}(M^4)$ with an almost complex structure ${{\cal J}}$ can be considered as a particular case of almost complex manifolds $X^6$ discussed in section 2 in the context of the pseudo-holomorphic bundles and the HYM equations.

[**Pull-back of complex vector bundles from $M^4$ to ${{\cal T}}(M^4)$**]{}. Let $E$ be a rank $k$ complex vector bundle over $M^4$ and $A$ a connection one-form (gauge potential) on $E$ with the curvature $F=\diff A+A\wedge A$ (gauge field). Suppose that the gauge field $F$ satisfies the ASDYM equations (\[2.7\]). Bundles $E$ with such a connection $A$ are called anti-self-dual [@AHS]. Using the projection (\[3.2\]), we pull $E$ back to a bundle $\hat E:=\pi^*E$ over ${{\cal T}}(M^4)$. In 