Unfamiliar Aspects of B\"acklund Transformations and an Associated Degasperis-Procesi Equation

We summarize the results of our recent work on B"acklund transformations (BTs), particularly focusing on the relationship of BTs and infinitesimal symmetries. We present a BT for an associated Degasperis-Procesi (aDP) equation and its superposition principle, and investigate the solutions generated by application of this BT. Following our general methodology, we use the superposition principle of the BT to generate the infinitesimal symmetries of the aDP equation.