Existence of nonlocal Lie symmetries for the integrable maps

The integrable map corresponding to the linear equation, respectively the nonlinear equation, is denoted by

$$, respectively$$

. A nonlocal diffeomorphism is a diffeomorphism whose action is nonlocal, and a Lie group of such transformations is denoted by $\mathcal{G}$. Nonlocal symmetry group conjecture. The integrable map

$$, respectively$$

, admits a Lie group $\mathcal{G}$ of nonlocal diffeomorphisms that leave the map invariant and transform solutions into solutions. This conjecture proposes the existence of nonlocal Lie symmetries for the categorical discretizations; whether such groups exist for the stated maps remains open.