accordance with the definition of a pull-back, the connection ${\hat{A}}:=\pi^* A$ on $\hat E$ is flat along the fibres ${\mathbb C}P^1_x$ of the bundle (\[3.2\]) and we can set the components of ${\hat{A}}$ along the fibres equal to zero. Thus, restrictions of the smooth vector bundle $\hat E$ to fibres ${\mathbb C}P^1_x$ of projection $\pi$ are holomorphically trivial for each $x\in M^4$.

It was shown in [@AHS] that if the Weyl tensor of $(M^4, g)$ is anti-self-dual then the almost complex structure ${{\cal J}}={{\cal J}}_+$ on the twistor space ${{\cal T}}(M^4)$ is integrable and ${{\cal T}}(M^4)$ inherits the structure of a complex analytic 3-manifold. Furthermore, it was proven that an anti-self-dual bundle $E$ over anti-self-dual $M^4$ lifts to a holomorphic bundle $\hat E$ over complex ${{\cal T}}(M^4)$ defined by the equation ${\hat{F}}^{0,2}=0$, where ${\hat{F}}:=\pi^*F$ is the pull-back to $\hat E$ of an anti-self-dual (ASD) gauge field $F$ on $E$. In [@AHS] it was also mentioned in a remark that one can introduce a Hermitian metric on ${{\cal T}}(M^4)$ such that ${\hat{F}}$ will be orthogonal to the Hermitian form. However, the HYM equations were introduced later [@Don; @UY] in a different context.

[**Generalized twistor correspondence**]{}. The essence of the canonical twistor 