  After giving the most general formulation to date of the notion of
integrability for axially symmetric harmonic maps from R^3 into symmetric
spaces, we give a complete and rigorous proof that, subject to some mild
restrictions on the target, all such maps are integrable. Furthermore, we prove
that a variant of the inverse scattering method, called vesture (dressing) can
always be used to generate new solutions for the harmonic map equations
starting from any given solution. In particular we show that the problem of
finding N-solitonic harmonic maps into a noncompact Grassmann manifold
SU(p,q)/S(U(p) x U(q)) is completely reducible via the vesture (dressing)
method to a problem in linear algebra which we prove is solvable in general. We
illustrate this method by explicitly computing a 1-solitonic harmonic map for
the two cases (p = 1, q = 1) and (p = 2, q = 1); and we show that the family of
solutions obtained in each case contains respectively the Kerr family of
solutions to the Einstein vacuum equations, and the Kerr-Newman family of
solutions to the Einstein-Maxwell equations.
