We derive a new four-dimensional partial differential equation with the isospectral Lax representation by shrinking the symmetry algebra of the reduced quasi-classical self-dual Yang–Mills equation and applying the technique of twisted extensions to the obtained Lie algebra. Then we find a recursion operator for symmetries of the new equation and construct a Bäcklund transformation between this equation and the four-dimensional Martínez Alonso–Shabat equation. Finally, we construct extensions of the integrable hierarchies associated to the hyper-CR equation for Einstein–Weyl structures, the reduced quasi-classical self-dual Yang–Mills equation, the four-dimensional universal hierarchy equation, and the four-dimensional Martínez Alonso–Shabat equation.

通过缩小缩小的拟经典自对偶Yang-Mills方程的对称代数并将扭曲扩展的技术应用于所获得的Lie代数，我们导出了一个具有等光谱Lax表示的新的四维偏微分方程。然后，我们为新方程的对称性找到一个递归算子，并在该方程和三维MartínezAlonso–Shabat方程之间建立一个Bäcklund变换。最后，我们构造了与爱因斯坦-魏尔结构的超CR方程，简化的拟经典自对偶杨-米尔斯方程，四维通用层次方程和四维马丁涅斯·阿隆索相关的可积层级的扩展–Shabat方程。