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)

(±1,0,0)

${\kappa }_4^{\left(a\right)}$

2

(0,±1,0)

${\kappa }_4^{\left(a\right)}$

2

$\left(\pm \frac{1}{\sqrt{2}},\pm \frac{1}{\sqrt{2}},0\right)$

$\frac{3{\kappa }_4^{\left(a\right)}+2}{4}$

4

(0,0,±1)

-2

2

$\left(0,\pm \sqrt{\frac{-2}{{\kappa }_4^{\left(a\right)}-2}},\pm \sqrt{\frac{{\kappa }_4^{\left(a\right)}}{{\kappa }_4^{\left(a\right)}-2}}\right)$

$\frac{2{\kappa }_4^{\left(a\right)}}{2-{\kappa }_4^{\left(a\right)}}$

4

${\kappa }_4^{\left(a\right)}\le 0$

$\left(\pm \sqrt{\frac{-2}{{\kappa }_4^{\left(a\right)}-2}},0,\pm \sqrt{\frac{{\kappa }_4^{\left(a\right)}}{{\kappa }_4^{\left(a\right)}-2}}\right)$

$\frac{2{\kappa }_4^{\left(a\right)}}{2-{\kappa }_4^{\left(a\right)}}$

4

${\kappa }_4^{\left(a\right)}\le 0$

$\left(\pm \sqrt{\frac{-4}{3{\kappa }_4^{\left(a\right)}-6}},\pm \sqrt{\frac{-4}{3{\kappa }_4^{\left(a\right)}-6}},\pm \sqrt{\frac{3{\kappa }_4^{\left(a\right)}+2}{3{\kappa }_4^{\left(a\right)}-6}}\right)$

$\frac{2(3{\kappa }_4^{\left(a\right)}+2)}{3(2-{\kappa }_4^{\left(a\right)})}$

8

${\kappa }_4^{\left(a\right)}\le -\frac{2}{3}$