 Solution

Put $w=V_{11}<0$. At $M=(\pi /a,\pi /a)$, the free energy is $2E_X$. The four cyclic combinations in the supplied secular equation give the four-plane-wave splitting at a square-zone cornerFor example, the normalized coefficient vectors in the source's cyclic basis are $(1,1,1,1)/2$, $(1,0,-1,0)/\sqrt{2}$, $(0,1,0,-1)/\sqrt{2}$, and $(1,-1,1,-1)/2$. Because $u<w<0$, these are ordered as written: a lower singlet, a central doublet and an upper singlet. The sketch puts the $X$ and $M$ energies on one common scale; its representative negative couplings are stated in the caption.