Nonlocal symmetries of the Degasperis–Procesi equation

We study nonlocal symmetries of the Degasperis–Procesi equation, which are shown to be closely related to its integrable structure. First, applying the Hamiltonian operator to the gradients of the spectral parameter, we construct nonlocal symmetries of the Kaup–Kupershmidt equation. Next, we show that the nonlocal symmetries can be prolonged to local symmetries for a prolonged system by introducing new dependent variables. Finally, applying the Liouville transformation relating the Degasperis–Procesi and Kaup–Kupershmidt hierarchies, we obtain the corresponding nonlocal symmetries of the Degasperis–Procesi equation.