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The correspondence between different versions of the Gauss-Weingarten equation is investigated.The compatibility condition for one version of the Gauss-Weingarten equation gives the Gauss-Mainardi-Codazzi system. A deformation of the surface is postulated which takes the same formas the original system but contains an evolution parameter. The compatibility condition of this newaugmented system gives the deformed Gauss-Mainardi-Codazzi system. A Lax representationin terms of a spectral parameter associated with the deformed system is established. Severalimportant examples of integrable equations based on the deformed system are then obtained. Itis shown that the Gauss-Mainardi-Codazzi system can be obtained as a type of reduction of theself-dual Yang-Mills equations

Deformation of Surfaces in Three-Dimensional Space Induced by Means of Integrable Systems