close to $(\omega,q_{-})=(0,0)$). Otherwise, if the curvature is convex (Hessian is positive definite) or the origin is a saddle-point (Hessian is indefinite), the plane wave is unstable. The curvature is characterized by the real eigenvalues of the symmetric Hessian matrix. The analysis of the eigenvalues of the Hessian matrix leads to the following condition for the stability of the plane wave, which is the condition for the negativeness of the eigenvalues of $H$: $$\left(\cos^{2}{k_{\tau}}-\sin^{2}{k_{-}}\right)\left(R\cos{k_{\tau}}+a^{2}\right)-R\cos{k_{\tau}}>0.$$ Using the amplitude relation (\[eq: SLpsolRe\]), the bifurcation is described by a 3rd order polynomial in $a^{2}$: $$0=a^{6}-\frac{5}{2}\alpha a^{4}+\left(2\alpha^{2}-\frac{R^{2}}{2}(1+2\sin^{2}{k_{-}}\right)a^{2}-\frac{\alpha^{3}}{2}+\frac{R^{2}\alpha}{2}(1+\sin^{2}{k_{-}}).\label{eq: SL 2-torus general neutral stability curve}$$ Solving the corresponding to (\[eq: SL 2-torus general neutral stability curve\]) equation for $a^{2}$ gives the neutral stability curve for an arbitrary plane wave with $|k_{-}|\in[0,\frac{\pi}{2}]$. The analytical solution can be found by using Cardano’s method, but is not written here for brevity. In the particular case $k_{-}=0$, the neutral stability curve can be simply expressed as $$a_{M}^{2}=\frac{1}{4}\left(3\alpha+\sqrt{\alpha^{2}+8C^{2}}\right),\label{eq: SL 2-torus Eckhaus curve}$$ which coincides with the result obtained in [@Yanchuk2008a] for the ring of coupled oscillators without delay (and also for a ring with large delay). By substituting (\[eq: SLpsolRe\]) into (\[eq: SL 2-torus general neutral stability curve\]), one can obtain 