# Table 1 List of all 26 solutions to Equation ( 7) for${\mathbf{\kappa }}_{\mathbf{4}}^{\mathbf{\left(}\mathbf{a}\mathbf{\right)}}\mathbf{\ne }\mathbf{2}$(second column), corresponding value of${\mathbf{\kappa }}_{\mathbf{4}}^{\mathbf{\left(}\mathbf{y}\mathbf{\right)}}$(third column) and condition for the solution g to be real-valued (last column)
Separating gT=(g 1,g 2,g 3) ${\kappa }_{4}^{\left(y\right)}$ Number of solutions Solution in$\mathbb{R}$
yes (±1,0,0) ${\kappa }_{4}^{\left(a\right)}$ 2
yes (0,±1,0) ${\kappa }_{4}^{\left(a\right)}$ 2
no $\left(±\frac{1}{\sqrt{2}},±\frac{1}{\sqrt{2}},0\right)$ $\frac{3{\kappa }_{4}^{\left(a\right)}+2}{4}$ 4
no $\left(0,±\sqrt{\frac{-2}{{\kappa }_{4}^{\left(a\right)}-2}},±\sqrt{\frac{{\kappa }_{4}^{\left(a\right)}}{{\kappa }_{4}^{\left(a\right)}-2}}\right)$ $\frac{2{\kappa }_{4}^{\left(a\right)}}{2-{\kappa }_{4}^{\left(a\right)}}$ 4 ${\kappa }_{4}^{\left(a\right)}\le 0$
no $\left(±\sqrt{\frac{-2}{{\kappa }_{4}^{\left(a\right)}-2}},0,±\sqrt{\frac{{\kappa }_{4}^{\left(a\right)}}{{\kappa }_{4}^{\left(a\right)}-2}}\right)$ $\frac{2{\kappa }_{4}^{\left(a\right)}}{2-{\kappa }_{4}^{\left(a\right)}}$ 4 ${\kappa }_{4}^{\left(a\right)}\le 0$
no $\left(±\sqrt{\frac{-4}{3{\kappa }_{4}^{\left(a\right)}-6}},±\sqrt{\frac{-4}{3{\kappa }_{4}^{\left(a\right)}-6}},±\sqrt{\frac{3{\kappa }_{4}^{\left(a\right)}+2}{3{\kappa }_{4}^{\left(a\right)}-6}}\right)$ $\frac{2\left(3{\kappa }_{4}^{\left(a\right)}+2\right)}{3\left(2-{\kappa }_{4}^{\left(a\right)}\right)}$ 8 ${\kappa }_{4}^{\left(a\right)}\le -\frac{2}{3}$