We derive a new four-dimensional partial differential equation with the isospectral Lax representation by shrinking the symmetry algebra of the reduced qua\-si-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 B{\"a}cklund transformation between this equation and the four-dimensional Mart{\'{\i}}nez Alonso--Shabat equation. Finally, we discuss integrability properties of the Lagrangian extension of the obtained equation.