Conjugate chains of discrete symmetries in $(1+2)$ nonlinear equations

We construct dressing chains of discrete symmetries for the Kadomtsev–Petviashvili equations and show that these equations admit two types of chains, which we call conjugate. We discuss the scheme of constructing dressing chains for the Boiti–Leon–Pempinelli equations. We find a nonsingular solution of these equations that is exponentially localized along some directions in the dissipative plane and is rational along other directions.