2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/67400We introduce a general notion of twistorial map and classify twistorial harmonic morphisms with one-dimensional fibres from self-dual four-manifolds. Such maps can be characterised as those which pull back Abelian monopoles to self-dual connections. In fact, the constructions involve solving a generalised monopole equation, and also the Beltrami fields equation of hydrodynamics, and lead to constructions of self-dual metrics.39 pages, LaTeX 2e, using fancyheadings sty. Some minor improvements made. To appear in Q. J. MathDifferential Geometry58E20, 53C43text