Symmetric reduction of the vectorial fundamental transformation: application to the Darboux-Egorov equations

The vectorial fundamental transformation for the Darboux equations is reduc ed to the symmetric case. This is combined with the orthogonal reduction of Lame type to obtain reductions of the vectorial Ribaucour transformations to Egorov systems. We also show that a permutability property holds for all these transformations. As an example we apply these transformations to the Cartesian background.