B\"acklund loop algebras for compact and non-compact nonlinear spin models in $(2+1)$ dimensions

The B"acklund problem is solved for both the compact and noncompact versions of the Ishimori (2+1)-dimensional nonlinear spin model. In particular, a realization of the arising B"acklund algebra in the form of an infinite-dimensional loop Lie algebra of the Ka\v{c}–Moody type is provided.