Lagrangian extensions of multi-dimensional integrable equations. I. The five-dimensional Martínez Alonso–Shabat equation

We study a Lagrangian extension of the 5d Martínez Alonso–Shabat equation \(\mathcal {E}\)$$\begin{aligned} u_{yz}=u_{tx}+u_y\,u_{xs}-u_x\,u_{ys} \end{aligned}$$that coincides with cotangent {T}^{*}\mathcal {E}\) to latter. describe Lie algebra structure its symmetries (which happens be quite nontrivial and is described in terms deformations) construct two families recursion operators for symmetries. Each family depends on parameters. prove that all from first are hereditary, but not compatible sense Nijenhuis bracket. also new parametric Lax pairs depend higher-order derivatives unknown functions.