any such pseudo-holomorphic bundle $\hat E\to{{\cal T}}(M^4)$ corresponds to a solution $A$ of the ASDYM equations on $M^4$.

Kähler geometry on twistor spaces of $S^4$ and $B_4$
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Here, as $M^4$ we consider the four-sphere $S^4$ with a metric $g$ of constant positive curvature and the open four-ball $B_4$ with a metric $g$ of constant negative curvature. In the next section 5 we shall consider the projective plane ${\mathbb C}P^2$ with the Fubini-Study metric and the complex two-ball ${\mathbb C}B_2$ with the metric of constant negative holomorphic sectional curvature. All these spaces $M^4$ are homogeneous manifolds (coset spaces) as well as their twistor spaces ${{\cal T}}(M^4)$. Although the geometry of these spaces is well-known, we describe it here by using local coordinates for fixing our notation. We also need this for self-consistency and further applications.

[**Manifolds $S^4$ and $B_4$ as coset spaces**]{}. Let us consider the group Sp(2) as a subgroup of SU(4) and the group Sp(1,1) as a subgroup of SU(2,2) defined as 4$\times$4 matrices $Q$ such that $$\label{4.1}
Q^\+\eta\,Q = Q\eta\, Q^\+=\eta{{\qquad{\rm with}\qquad}}\eta =\rm{diag}({\bf 1}_2, \ve{\bf 1}_2)\ ,$$ where $\ve =1$ for Sp(2)$\,\subset\,$SU(4) and $\ve =-1$ for Sp(1,1)$\,\subset\,$SU(2,2). We consider $S^4$ and $B_4$ as coset spaces $$\label{4.2}
S^4={\rm 