P. Bracken - The generalized Weierstrass system for mean curvature surfaces and the completely integrable sigma model

The integrability of a system which describes constant mean curvature surfaces by means of the generalized Weierstrass inducing formula is studied. This is carried out by using a specific transformation which reduces the initial system to the completely integrable two-dimensional Euclidean sigma model. A new linear spectral problem equivalent to the generalized Weierstrass system is derived via the method of differential constraints. Furthermore the Auto-Bäcklund transformation for the generalized Weierstrass system can also be constructed. The permutability theorem for this Auto-Bäcklund transformation is formulated. New classes of non-splitting multi-soliton solutions are obtained by exploiting this Auto-B"acklund Transformation. A technique for reduction of generalized Weierstrass system to decoupled linear equations by subjecting it to certain differential constraints is presented as well.