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 form as the original system but contains an evolution parameter. The compatibility condition of this new augmented system gives the deformed Gauss-Mainardi-Codazzi system. A Lax representation in terms of a spectral parameter associated with the deformed system is established. Several important examples of integrable equations based on the deformed system are then obtained. It is shown that the Gauss-Mainardi-Codazzi system can be obtained as a type of reduction of the self-dual Yang-Mills equations.