included in the case 1 or 2 (the case 1 or 3) as an additional requirement.

The case of 5 for NH is not further discussed because it does not supply no useful linear relations with respect $a_{1,+,-}$ and $b_{1,+,-}$ and the cases of 4 and 5 for IH are irrelevant.

\[sec:3\]Maximal CP Violation
=============================

In this section, we would like to find appropriate conditions on $a_{1,2,3}$ and $b_{1,2,3}$, which are similar to Eqs.(\[Eq:Maximal-CP1\]) and (\[Eq:Maximal-CP2\]), to induce maximal CP violation. From the discussions in Sec.\[sec:2\], we find several such candidates in both NH and IH. We choose the phase to be ${e^{-i{\delta _{CP}}}}$ appearing in the equations as much the same way as in Eq.(\[Eq:M\_nu for 23\]). The results are summarized in TABLE \[Tab:NH\] for NH and TABLE \[Tab:IH\] for IH that show which Dirac neutrino masses are real or imaginary. In these tables, the real or imaginary Dirac neutrino masses give maximal CP violation through the relevant constraint(s).

   case                             relevant constraint for $\delta _{CP}=\pm \pi/2$                                  real        imaginary
  ------ ------------------------------------------------------------------------------------------------------- -------------- --------------
    1                     ${a_ - }{e^{-i{\delta _{CP}}}} =  - {{{s_{13}}}}{a_ + }/{{{t_{12}}}}$                      $a_-$          $a_+$
    2                     ${b_ - }{e^{-i{\delta _{CP}}}} =  - {{{s_{13}}}}{b_ + }/{{{t_{12}}}}$                      $b_-$          $b_+$
    3     ${a_1}{e^{-i{\delta _{CP}}}} 