The Manakov-Zakharov-Ward model as an integrable decoupling limit of the membrane

A novel decoupling limit of the membrane is proposed, leading to the $(1+2)$-dimensional classically integrable model originally introduced by Manakov, Zakharov, and Ward. This limit can be interpreted as either a non-relativistic or large-wrapping regime of a membrane propagating in a toy background of the form $\mathbb{R}_t \times T^2 \times G$, where $G$ is a Lie group and the geometry is supported by a four-form flux. We demonstrate that such toy backgrounds can arise from consistent eleven-dimensional supergravity solutions, exemplified by the uplift of the pure NSNS AdS$_3 \times$ S$^3 \times$ T$^4$ background.