In this paper, we present a method for obtaining solutions to the Calapso and Zoomeron equations using Ribaucour transformations. We provide explicit formulas for the solutions of these equations. By applying this method to the generalized cylinder, we provide new solutions of these equations, which are determined by a free function and by two other functions, each one defined in a given variable. As a consequence, we provide new solutions for the Davey-Stewartson III equation.

In this paper, we consider a method of constructing flat surfaces based on Ribaucour transformations in S3. By applying the theory to the flat torus, we obtain a family of complete flat surfaces in S3 which is determined by several parameters. We provide explicit examples.

In this paper we consider M a fixed hypersurface in Euclidean space and we introduce two types of spaces relative to M, of type I and type II. We observe that when M is a hyperplane, the two geometries coincides with the isotropic geometry. By applying the theory to a Dupin hypersurface M, we define a relative Dupin hypersurface M of type I and type II , we provide necessary and sufficient conditions for a relative hypersurface M to be relative Dupin parameterized by relative lines of curvature, in both spaces. Moreover, we provides a relationship between the Dupin hypersurfaces locally associated to M by a Ribaucour transformation and the type II Dupin hypersurfaces relative M. We provide explicit examples of the Dupin hypersurface relative to a hyperplane, torus, S1  x  Rn-1 and  S2 x  Rn-2, in both spaces.

In this paper, we consider a method of constructing isothermic surfaces in S3 based on Ribaucour transformations. By applying the theory to the flat torus, we obtain two family of complete isothermic surfaces in S3. One four-parameter family of complete isothermic surfaces that contains n-bubble surfaces inside and outside of the torus. We also get another four-parameter of complete isothermic surfaces which are Dupin surfaces. As aplication we obtain explicit solutions of the Calapso equation.