On the construction of monopoles

We show that any self-dual SU (2) monopole may be constructed either by Ward's twistor method, or Nahm's use of the ADHM construction. The common factor in both approaches is an algebraic curve whose Jacobian is used to linearize the non-linear ordinary differential equations which arise in Nahm's method. We derive the non-singularity condition for the monopole in terms of this curve and apply the result to prove the regularity of axially symmetric solutions. © 1983 Springer-Verlag.