approach is to establish a correspondence between four-dimensional space-time $M^4$ and [*complex*]{} twistor space ${{\cal T}}(M^4)$ of $M^4$. Using this correspondence, one transfers data given on $M^4$ to data on ${{\cal T}}(M^4)$ and vice versa. In twistor theory one considers [*holomorphic*]{} objects $h$ on ${{\cal T}}(M^4)$ (Čech cohomology classes, holomorphic vector bundles etc.) and transforms them to objects $f$ on $M^4$ which are constrained by some differential equations [@Penrose; @Ward77; @AHS; @Wells]. Thus, the main idea of twistor theory is to encode solutions of some differential equations on $M^4$ in holomorphic data on the complex twistor space ${{\cal T}}(M^4)$ of $M^4$. In particular, solutions of the ASDYM equations on manifolds $M^4$ with the ASD Weyl tensor correspond to holomorphic vector bundles $\hat E$ over ${{\cal T}}(M^4)$. However, in Donaldson theory [@DK] one considers the ASDYM equations on manifolds $M^4$ whose Weyl tensor is not restricted and it is desirable to have a twistor description of this generic case.[^3].

In [@Popov] it has been shown that the vortex equations on a compact Riemann surface $\Sigma$ are equivalent to the ASDYM equations on $\Sigma\times{\mathbb C}P^1$ and to the Hermitian-Yang-Mills equations on the twistor space ${{\cal T}}(M^4)$ of $M^4=\Sigma\times{\mathbb C}P^1$. In the general 