Backlund Transformations for Darboux Integrable Differential Systems: Examples and Applications

In the article arXiv:1108.5443 we established a general group-theoretical approach to the construction of B"acklund transformations. We then showed how this construction can be applied to construct B"acklund transformation between equations which are Darboux integrable. Here we give a number of detailed examples and new applications which demonstrate the theory. In particular our final example demonstrates how our group theoretical approach produces all the B"acklund transformations in arXiv:0707.4408. We also prove, using group methods, a non-existence theorem for B"acklund transformatons which disagrees with part 2 of Theorem 1 in arXiv:0707.4408.