On Ricci solitons and twistorial harmonic morphisms

We study the soliton flow on the domain of a twistorial harmonic morphism between Riemannian manifolds of dimensions four and three. Assuming real-analyticity, we prove that, for the Gibbons-Hawking construction, any soliton flow is uniquely determined by its restriction to any local section of the corresponding harmonic morphism. For the Beltrami fields construction, we identify a contour integral whose vanishing characterises the trivial soliton flows.