and Eq.(\[Eq:12 for NHseesaw\]) for $\theta_{12}$, we, respectively, obtain ${a_ + }{a_ - } =  - \left( {{b_ + } + {t_{13}}{e^{ - i{\delta _{CP}}}}{b_1}} \right){b_ - }$ and ${b_1} = \frac{{{t_{12}}}}{{{c_{13}}}}{b_ - } + {t_{13}}{b_ + }{e^{i{\delta _{CP}}}}$, which turn out to satisfy (\[Eq:13 for NHseesaw\]) for $\theta_{13}$. Inserting the expression of $b_1$ into those of $a_ - ^2$ and $a_+a_-$, finally, gives a simpler relation ${a_ - } =  - \frac{{{s_{13}}}}{{{t_{12}}}}{a_ + }{e^{i{\delta _{CP}}}}$. We obtain that ${b_1} = \frac{{{t_{12}}}}{{{c_{13}}}}{b_ - } + {t_{13}}{b_ + }{e^{i{\delta _{CP}}}}$ and ${a_ - } =  - \frac{{{s_{13}}}}{{{t_{12}}}}{a_ + }{e^{i{\delta _{CP}}}}$ as useful relations together with $a_-=0$.

2.  $a_+=0$: It is readily recognized that no simple linear relations are deduced from the equations and $a_{1,-}$ and $b_{1,+,-}$ should satisfy $a_ - ^2 + b_ - ^2 = \frac{1}{{{t_{12}}{c_{13}}}}\left( {{a_1}{a_ - } + {b_1}{b_ - }} \right)$ from Eq.(\[Eq:m1=0 for seesaw\]), ${b_ + }{b_ - } =  - {t_{13}}{e^{ - i{\delta _{CP}}}}\left( {{a_1}{a_ - } + {b_1}{b_ - }} \right)$ from Eq.(\[Eq:23 for NHseesaw\]), $a_1^2 + b_1^2 - t_{12}^2\left( {a_ - ^2 + b_ - ^2} \right) = {t_{13}}{b_1}{b_ + }{e^{i{\delta _{CP}}}}$ from Eq.(\[Eq:12 for NHseesaw\]) and $c_{13}^2\left( {a_1^2 + b_1^2} \right) - 