Darboux transformations for the twisted so(p, q) system and local isometric immersion of space forms

For the n -dimensional integrable system with a twisted reduction, Darboux transformations given by Darboux matrices of degree two are constructed explicitly. These Darboux transformations are applied to the local isometric immersion of space forms with flat normal bundle and linearly-independent curvature normals to give the explicit expression of the position vector. Some examples are given from the trivial solutions and standard imbedding .