$F$, respectively.

For the pull-back ${\hat{F}}^{\pm}:=\pi^*F^{\pm}$ of the two-forms (\[3.8\]) and (\[3.9\]) on $M^4$ to ${{\cal T}}(M^4)$ we obtain $$\label{3.10}
{\hat{F}}^+=\sfrac{1}{2}\,({\hat{F}}_{1\1}+{\hat{F}}_{2\2})(\o^{1\1}+\o^{2\2})+
{\hat{F}}_{12}\o^{12}+{\hat{F}}_{\1\2}\o^{\1\2}\ ,$$ $$\label{3.11}
{\hat{F}}^-=\sfrac{1}{2}\,({\hat{F}}_{1\1}-{\hat{F}}_{2\2})(\o^{1\1}-\o^{2\2})+
{\hat{F}}_{1\2}\o^{1\2}+{\hat{F}}_{2\1}\o^{2\1}\ ,$$ where $\o^{1\1}:=\o^{1}\wedge\o^{\1},\ \o^{12}:=\o^{1}\wedge\o^{2}$ etc. Note that $$\begin{aligned}
\o^{12}=\frac{1}{1{+}\zeta{\bar{\zeta}}}\,[\th^{12}+\zeta (\th^{1\1}+\th^{2\2})
+\zeta^2\th^{\1\2}]\ ,\quad \o^{1\2}=\th^{1\2}\ ,\nonumber\\
\o^{\1\2}=\frac{1}{1{+}\zeta{\bar{\zeta}}}\,[\th^{\1\2}-{\bar{\zeta}}(\th^{1\1}+\th^{2\2})
+{\bar{\zeta}}^2\th^{12}]\ ,\quad \o^{2\1}=\th^{2\1}\ ,\nonumber\\
\o^{1\1}{+}\o^{2\2}=\frac{1}{1{+}\zeta{\bar{\zeta}}}\,[(1{-}\zeta{\bar{\zeta}})
(\th^{1\1}{+}\th^{2\2}){+}2\zeta\th^{\1\2}{-}2{\bar{\zeta}}\th^{12}]\ ,\quad
\o^{1\1}{-}\o^{2\2}=\th^{1\1}{-}\th^{2\2}\ ,\label{3.12}\end{aligned}$$ as one can easily derive from (\[3.5\]). Also we have $$\begin{aligned}
{\hat{F}}_{12}=\frac{1}{1{+}\zeta{\bar{\zeta}}}\,[F_{12}+{\bar{\zeta}}(F_{1\1}+F_{2\2})
+{\bar{\zeta}}^2F_{\1\2}]\ ,\quad {\hat{F}}_{1\2}=F_{1\2}\ ,\nonumber\\
{\hat{F}}_{\1\2}=\frac{1}{1{+}\zeta{\bar{\zeta}}}\,[F_{\1\2}-\zeta (F_{1\1}+F_{2\2})
+\zeta^2F_{12}]\ ,\quad {\hat{F}}_{2\1}=F_{2\1}\ ,\nonumber\\
{\hat{F}}_{1\1}{+}{\hat{F}}_{2\2}=\frac{1}{1{+}\zeta{\bar{\zeta}}}\,[(1{-}\zeta{\bar{\zeta}})
(F_{1\1}{+}F_{2\2}){-}2\zeta F_{12}{+}2{\bar{\zeta}}F_{\1\2}]\ ,\quad
{\hat{F}}_{1\1}{-}{\hat{F}}_{2\2}=F_{1\1}{-}F_{2\2}\ ,\label{3.13}\end{aligned}$$ and by construction $$\label{3.14}
{\hat{F}}_{i\3}={\hat{F}}_{i3}=0\und\rm{h.c.}$$ for $i=1,2,3$.

Using (\[3.5\]), we can introduce on ${{\cal T}}(M^4)$ an almost Hermitian form $$\label{3.15}
\o =\sfrac{\im}{2}\,\left(\o^1\wedge\o^{\1}+\o^2\wedge\o^{\2}+
\ve\o^3\wedge\o^{\3}\right)\ ,$$ where $\ve=\pm 1$.[^6] Then for ${\hat{F}}={\hat{F}}^-$ from (\[3.11\]) it follows that $$\label{3.16}
{\hat{F}}^{0,2}=0$$ and $$\label{3.17}
\o\,\lrc\,{\hat{F}}=0\ .$$ Thus, anti-self-dual gauge fields $F=F^-$ on the vector bundle $E\to M^4$ are pulled back to the gauge fields ${\hat{F}}$ on the vector bundle $\hat E$ over the twistor space ${{\cal T}}(M^4)$ which satisfy the Hermitian-Yang-Mills equations (\[3.16\]), (\[3.17\]) on ${{\cal T}}(M^4)$ without demanding integrability of an almost complex structure (\[3.6\]). In its turn, such gauge fields ${\hat{F}}$ define a pseudo-holomorphic structure on the vector bundle $\hat E$ which is holomorphically trivial on ${\mathbb C}P^1_x\hra
{{\cal T}}(M^4)$ for each $x\in M^4$. Conversely, 