Symmetry Reduction of Exterior Differential Systems and Backlund   Transformations for PDE in the Plane
  We approach the construction of Backlund transformations for Darboux integrable hyperbolic partial differential equations in the plane through the reduction of exterior differential systems. For example it is shown that all the Backlund transformations in arXiv:0707.4408v2 can be constructed using symmetry reduction. 
I. M. Anderson, M. E. Fels
math.DG