The Weierstrass-Enneper system for constant mean curvature surfaces and the completely integrable sigma model

The integrability of a system which describes constant mean curvature surfa ces by means of the adapted Weierstrass-Enneper 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 nonli near sigma model. Through the use of the apparatus of differential forms an d Cartan theory of systems in involution, it is demonstrated that the gener al analytic solutions of both systems possess the same degree of freedom. F urthermore, a new linear spectral problem equivalent to the initial Weierst rass-Enneper system is derived via the method of differential constraints. A new procedure for constructing solutions to this system is proposed and i llustrated by several elementary examples, including a multi-soliton soluti on. (C) 1999 American Institute of Physics. [S0022-2488(99)03307-1].