d\tau dx^{3}\sqrt{-g}F\left(  \psi\right)  N^{2}R=\int d\tau dx^{3}\sqrt{-g}\left(  -\frac{R}{2}\right)  .$$ On the other hand the second term in Eq.(\[ICLN\]) gives, after integration by parts:

$$\begin{aligned}
\label{ICLN1}
\int d\tau dx^{3}\sqrt{-g}\left[ -6F\left( \psi \right) N\Delta _{2}N\right]
&=&\int d\tau dx^{3}\sqrt{-g}\left( -6\frac{F}{\sqrt{-2F}} N_{;ij}g^{ij}\right) \nonumber\\
&=&\int d\tau dx^{3}\sqrt{-g}\left[ -6\frac{F}{\sqrt{-2F}}\frac{1}{\sqrt{-g}}\left( \sqrt{-g}g^{ij}N_{,k}\right) _{,j}\right] \nonumber\\
&=&\int d\tau dx^{3}\left[ -6\frac{F}{\sqrt{-2F}}\left( \sqrt{-g}
g^{ij}N_{,k}\right) _{,j}\right] \nonumber\\
&=&\int d\tau dx^{3}\sqrt{-g}\left( 3\frac{F_{\psi }}{\sqrt{-2F}}\psi
_{;j}N_{;i}g^{ij}\right) \nonumber \\
&=&
\int d\tau dx^{3}\sqrt{-g}\left[ 3\frac{F_{\psi }^{2}}{\left( -2F\right)
^{2}}\psi _{;i}\psi _{;j}g^{ij}\right] \;.\end{aligned}$$

The third term provides: $$\frac{\varepsilon}{2}N^{2}\Delta_{1}\psi=\frac{\varepsilon}{4F}\psi_{;i}\psi_{;j}g^{ij}\,.$$ Finally, collecting all terms and inserting them into Eq.(\[ICLN\]), the action is written as

$$\begin{aligned}
S_{NM} &=&
\int d\tau dx^{3}\sqrt{-g}\left[ -\frac{R}{2}+3\frac{F_{\psi }^{2}}{4F^{2}
}\psi _{;i}\psi _{;j}g^{ij}-\frac{\varepsilon }{4F}\psi _{;i}\psi
_{;j}g^{ij}-\frac{\bar{V}\left( \psi \right) }{4F^{2}}\right] \nonumber\\
&=&
\int d\tau dx^{3}\sqrt{-g}\left[ -\frac{R}{2}+\frac{\varepsilon }{2}\left( \frac{3\varepsilon F_{\psi }^{2}-F}{2F^{2}}\right) \psi _{;i}\psi
_{;j}g^{ij}-\frac{\bar{V}\left( \psi \right) }{4F^{2}}\right] \;.
\label{CLN.12.5}\end{aligned}$$

Interestingly, introducing the scalar field $\Psi$ with the requirement: $$d\Psi=\sqrt{\left(  \frac{3\varepsilon F_{\psi}^{2}-F}{2F^{2}}\right)  }d\psi\label{CLN.12.6}$$ the action of Eq.(\[CLN.12.5\]) can be written as follows $$S_{NM}=\int d\tau dx^{3}\sqrt{-g}\left[  -\frac{R}{2}+\frac{\varepsilon}{2}\Psi_{;i}\Psi_{;j}g^{ij}-\frac{\bar{V}\left(  \Psi\right)  }{4F\left(
\Psi\right)  ^{2}}\right]  \;.\label{CLN.12.7}$$ We conclude that the scalar field $\Psi$ is minimally coupled (modulus a constant) to the gravitational field. In other words, we find that to every non-minimally coupled scalar field, we may associate a unique minimally coupled scalar field in a conformally related space 