s_{13}^2{e^{2i{\delta _{CP}}}}b_ + ^2 = \left( {c_{13}^2 - s_{13}^2} \right)t_{12}^2\left( {a_ - ^2 + b_ - ^2} \right)$ from Eq.(\[Eq:13 for NHseesaw\]).

3.  $a_-=0$: From Eq.(\[Eq:m1=0 for seesaw\]), ${b_ - } = \frac{1}{{{t_{12}}{c_{13}}}}{b_1}$ is required to have $m_1 = 0$. From Eq.(\[Eq:23 for NHseesaw\]) and Eq.(\[Eq:12 for NHseesaw\]), we, respectively, obtain ${b_ + } =  - {t_{13}}{b_1}{e^{ - i{\delta _{CP}}}}$ and ${a_1} = {t_{13}}{a_ + }e^{i{\delta _{CP}}}$, which turn out to satisfy (\[Eq:13 for NHseesaw\]) for $\theta_{13}$. We obtain that ${b_ + } + {t_{13}}{b_1}{e^{ - i{\delta _{CP}}}}=0$, ${b_ - } = \frac{1}{{{t_{12}}{c_{13}}}}{b_1}$ and ${a_1} = {t_{13}}{a_ + }e^{i{\delta _{CP}}}$ together with $a_1=0$.

For IH, the combined use of Eqs.(\[Eq:m3=0 for seesaw\]) and (\[Eq:13 for IHseesaw\]) yields ${a_1}{a_ + } + {b_1}{b_ + } = -{t_{13}}\left( {a_1^2 + b_1^2} \right){e^{ - i{\delta _{CP}}}}$ for Eq.(\[Eq:13 for IHseesaw\]) giving $$\left( {{a_ + } + {t_{13}}{a_1}{e^{ - i{\delta _{CP}}}}} \right){a_1} + \left( {{b_ + } + {t_{13}}{b_1}{e^{ - i{\delta _{CP}}}}} \right){b_1} = 0,
\label{Eq:13-simplerIH}$$ by which Eq.(\[Eq:m3=0 for seesaw\]) is further reduced to $$\left( {{a_ + } + {t_{13}}{a_1}{e^{ - i{\delta _{CP}}}}} \right){a_ + } + \left( {{b_ + } + {t_{13}}{b_1}{e^{ - i{\delta _{CP}}}}} \right){b_ + } = 0.
\label{Eq:m3=0-simplerIH}$$ Similarly, 