Geodesic
Twistors and $G$-structures
Izvestiya. Mathematics , Tome 40 (1993) no. 1, pp. 1-31

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The authors distinguish a class of twistor spaces $Z=P\times_GS$ that are associated, following Berard-Bergery and Ochiai, with $G$-structures $P$ on even-dimensional manifolds and connections in $P$. It is assumed that $S=G/H$ is a complex totally geodesic submanifold of the affine symmetric space $\operatorname{GL_{2n}}(\mathbf R)/\operatorname{GL_n}(\mathbf C)$. This class includes all the basic examples of twistor spaces fibered over an even-dimensional base. The integrability of the canonical almost complex structure $J_Z$ and the holomorphy of the canonical distribution $\mathscr H_Z$ in $Z$ are studied in terms of some natural $H$-structure with a connection on the manifold $Z$. Some examples are also treated. 
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D. V. Alekseevskii; M. M. Graev. Twistors and $G$-structures. Izvestiya. Mathematics , Tome 40 (1993) no. 1, pp. 1-31. http://geodesic.mathdoc.fr/item/IM2_1993_40_1_a0/