Subalgebra conjecture for nonlocal and classical Lie symmetries

Let $\mathcal{G}$ be the Lie group of nonlocal symmetry diffeomorphisms associated with the integrable map (respectively, ), and let and denote the corresponding continuous dynamical systems. Subalgebra conjecture. The Lie algebra of the generators of $\mathcal{G}$ contains a subalgebra isomorphic to the Lie algebra of the classical Lie point symmetries of the continuous dynamical system (respectively, ). This conjecture predicts a weaker correspondence between the symmetry algebras of categorically related continuous and discrete models, since their full Lie algebras need not be isomorphic; the source does not provide evidence establishing or resolving the claim.