Gauge Theoretical Method in Solving Zero-curvature Equations I. -- Application to the Static Einstein-Maxwell Equations with Magnetic Charge

The inverse scattering problem is applied to 2-dimensional partial differential equations called soliton equations such as the KdV equation and so on. It is also used to integrate the Einstein equations with axial symmetry. These inverse scattering problems look different. We show that they can be understood in a unified way. As an application to the Einstein equation, we find solutions of the Einstein-Maxwell equations with a magnetic charge.