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

Lagrangian extensions of multi-dimensional integrable equations. I. The five-dimensional Mart{\'{\i}}nez Alonso--Shabat equation