coset spaces (\[5.15\]). We again have fibrations (\[4.28\]) but with $M^4\cong{\mathbb C}P^2$ or $M^4\cong{\mathbb C}B_2$.

[**Kähler structure on twistor spaces ${{\cal T}}({\mathbb C}P^2)$ and ${{\cal T}}({\mathbb C}B_2)$**]{}. Consider a trivial complex vector bundle ${{\cal T}}(M^4)\times{\mathbb C}^4\to{{\cal T}}(M^4)$ with $M^4\cong{\mathbb C}P^2$ or ${\mathbb C}B_2$. A flat connection on this bundle is defined by formula $$\label{5.18}
\hat{{\cal A}}= \hat V^{-1}\diff\hat V =\hat G^{-1}{{\cal A}}\,\hat G +
\hat G^{-1}\diff\hat G =:
 \begin{pmatrix}2b&-\sfrac{\ve}{2\La}\,\hat \th^{\+}\\
\sfrac{1}{2\La}\hat\th&\hat B\end{pmatrix} \ ,$$ where $$\label{5.19}
 \hat\th = g^{\+}\th =\frac{1}{(1+\zeta{\bar{\zeta}})^{\frac{1}{2}}}
\begin{pmatrix}
\th^{\2}+\bar\zeta\th^1\\ \th^1-\zeta\th^{\2}
\end{pmatrix}
=:
\begin{pmatrix}
 \o^{\2}\\ \o^1
\end{pmatrix}\ ,\qquad
\hat\th^{\+}=\th^{\+}g=(\o^2\ \o^{\1})\ ,$$ $$\label{5.20}
 \hat B = g^{\+}B\,g +g^{\+}\diff\,g=:
\begin{pmatrix}
 \hat a_+& -\sfrac{1}{2R}\o^{\3}\\ \sfrac{1}{2R}\o^{3}&-\hat a_+
\end{pmatrix} -b\cdot{\bf 1}_2$$ with $$\label{5.21}
\hat a_+ =\frac{1}{1+\zeta{\bar{\zeta}}}\,\left\{ (1-\zeta{\bar{\zeta}})a_+ + {\bar{\zeta}}b_+ -
\zeta\bar b_+ + \frac{1}{2}({\bar{\zeta}}\diff\zeta -\zeta\diff{\bar{\zeta}})\right\}\ ,$$ $$\label{5.22}
\o^3 =\frac{2R}{1+\zeta{\bar{\zeta}}}\, \left(\diff\zeta  + b_+ -2\zeta a_+ +
\zeta^2\bar b_+\right)\ ,$$ and $b, a_+, b_+$ are given in (\[5.9\])-(\[5.12\]).

From flatness of $\hat{{\cal A}}$ we obtain $$\label{5.23}
\hat{{\cal F}}= \diff\hat {{\cal A}}+\hat{{\cal A}}\wedge\hat{{\cal A}}=
 \begin{pmatrix}
2\diff b -\frac{\ve}{4\La^2}\,\hat\th^{\+}\wedge\hat\th&
-\frac{\ve}{2\La}\,(\diff\hat\th^{\+}{+}\hat\th^{\+}\wedge
\hat B{-}2\hat\th^{\+}\wedge b)
\\[2mm]
\frac{1}{2\La}\,(\diff\hat\th{+}\hat B\wedge\hat \th{-}2b\wedge\hat\th)&
\hat F^+ -\diff b\cdot{\bf 1}_2 -\frac{\ve}{4\La^2}\,\hat\th\wedge\hat\th^{\+}
\end{pmatrix} =0$$ and therefore $$\label{5.24}
 f^-=\diff b = -\frac{\ve}{8\La^2}\,(\o^1\wedge\o^{\1}-\o^2\wedge\o^{\2})=
-\frac{\ve}{8\La^2}\,(\th^1\wedge\th^{\1}-\th^2\wedge\th^{\2})\ ,$$ $$\label{5.25}
\hat F^+ = -\frac{\ve}{8\La^2}\,
 \begin{pmatrix}
\o^1\wedge\o^{\1}+\o^2\wedge\o^{\2}& 2\o^{\1}\wedge\o^{\2}\\
-2\o^{1}\wedge\o^{2}&-\o^1\wedge\o^{\1}-\o^2\wedge\o^{\2}
 \end{pmatrix}$$ along with $$\label{5.26}
