A famous theorem of Donaldson describes a correspondence between $\mbox{SU}(2)$ monopolesover three-dimensional Euclidean space and maps from $\Bbb{CP}^1$ to itself. This paper generalises this tomonopoles with arbitrary gauge group $G,$ and new maps from $\Bbb{CP}^1$ to flag manifolds $G/H$ are produced.This is in line with a conjecture of Atiyah and Murray, following a similar result of Atiyah's on hyperbolicmonopoles.

Donaldson's approach, also followed by Hurtubise and Murray in previous proofs of manyimportant cases of the current result, depends on a description of monopoles in terms of a system of ordinarydifferential equations known as Nahm's equations. In contrast, our approach is more direct, and the bulk ofthe paper is concerned with describing the rational map associated to a particular framed monopole, viasolutions to a scattering equation (first introduced by Hitchin) along parallel lines.

A subsidiarysection of the paper analyses rational maps into flag manifolds, constructing canonical lifts into larger flagmanifolds, and into the (complexified) Lie group. The main result is dependent upon additional work in acompanion paper, ‘Construction of Euclidean monopoles’, to appear in the same journal, where a procedure isdescribed for recovering a monopole from its rational map.

1991 Mathematics Subject Classification:53C80, 58D27, 58E15, 58G11.