of $a_{1,+,-}$ and $b_{1,+,-}$, Eqs.(\[Eq:Maximal-CP1\]) and (\[Eq:Maximal-CP2 and ATM\]) can be expressed as ${\rm Re} \left( {{a_ + }{a_ - } + {b_ + }{b_ - }} \right) = 0$ and ${\rm Im} \left( {{a_1}{a_ - } + {b_1}{b_ - }} \right) = 0$.

So far, we have assumed that one of $a_{1,+,-}$ and $b_{1,+,-}$ vanishes but more general conclusion can be obtained without making any assumptions. It is known that the relations of $M_{e\tau} = -\sigma M_{e\mu}^\ast$ and $M_{\tau\tau} = M_{\mu\mu}^\ast$ supplemented by $M_{ee,\mu\tau}$=real lead to maximal CP violation as well as maximal atmospheric neutrino mixing [@MassTextureCPEarlier; @MassTextureCP]. In our point of view, it is understood that these relations serve as specific solutions to Eqs.(\[Eq:Maximal-CP1\]) and (\[Eq:Maximal-CP2 and ATM\]) [@Useful1]. In terms of $a_{1,+,-}$ and $b_{1,+,-}$, the solution consists of ${a_3} = -\sigma{a_2}^\ast$ and ${b_3} = -\sigma{b_2}^\ast$ supplemented by ${a_1} = {\rm real}$ and ${b_1} = {\rm real}$. The Dirac neutrino masses are uniquely determined to be: $$\begin{aligned}
m_D = 
\left(
  \begin{array}{cc}
    \sqrt{M_1}a_1  & \sqrt{M_2}b_1   \\
    \sqrt{M_1}a_2  & \sqrt{M_2}b_2   \\
    \sqrt{M_1}(-\sigma a^\ast_2)  & \sqrt{M_2}(-\sigma b^\ast_2)   \\
  \end{array}
\right),
\label{Eq:MaximalCPmD}\end{aligned}$$ where $a_1$ and $b_1$ are real. As in Ref.[@MassTextureCP; @FlavoredSym], if a unitary matrix $S$ is defined to be $$\begin{aligned}
