Pseudotwistors

We deal with the Hermitian Hurwitz pairs of signature (, s), + s = 5 + 4, | + 1 – s| = 2 + 4m;, m = 0, 1,.... We introduce the Hurwitz twistors for signature (3, 2) and its dual (1, 4) and we indicate the relationship between Hurwitz and Penrose twistors. The signatures (1, 8) and (7, 6) give rise to pseudotwistors and bitwistors, respectively. For pseudotwistors, we prove a counterpart of the Penrose theorem in the local version, on real analytic solutions of the related spinor equations versus harmonic forms, and in the semiglobal version, on holomorphic solutions of those equations versus Dolbeault cohomology groups. We prove an atomization theorem: There exist complex structures on isometric embeddings for the Hermitian Hurwitz pairs so that the embeddings are real parts of holomorphic mappings.