This is a joint work with Yutaro Kanata．
We investigate pseudospheres as an object of singularity theory. A pseudosphere is a surface in the Euclidean 3-space whose Gauss curvature is constantly equal to -1. By Hilberta's theorem, such a surface cannot be complete. In other words, its completion must have singularities. We describe how singularities of such a surface appear in terms of Chebyshev's net. We also describe ridge points and flecnodal points on pseudosphere. Ridge is the notion corresponding to A3-singularity of the distance square function and flecnodal corresponds to the notion corresponding to 4-point contact of a line with the surface. These notions are found in the context of singularity theory. Next we investigate how the singularities behave by Backlund transformation, which is an attractive map between pseudospheres. Backlund transformations create new pseudospheres from a known pseudosphere. As an application, we obtain a classification result for 2-soliton surfaces
which are pseudospheres obtained by a line applying Backlund transformation twice.