Isometric deformations of surfaces preserving the third fundamental form

We study the following problem: To what extend is a surface in the Euclidean space (mathbb{R}^{4}) determined by the third fundamental form? We prove the existence of families of surfaces in (mathbb{R}^{4}) which allow isometric deformations with isometric but not congruent Gaussian images. In particular, we provide a method which gives locally all surfaces in (mathbb{ R}^{4}) with conformal Gauss map that allow such deformations. As a consequence, we have a way for constructing non-spherical pseudoumbilical surfaces in (mathbb{R}^{4}.)