coupling function $F(\psi)$ in the gravitational field $\bar{g}_{ij}$ are the same with the field equations of the minimally coupled scalar field $\Psi$ for a conformal Lagrangian $L\left(  \tau,x^{k},\dot
{x}^{k}\right)  $ in the conformal metric $g_{ij}$ $=N^{-2}\bar{g}_{ij}$ where the conformal function is $N=\frac{1}{\sqrt{-2F\left(  \psi\right)  }}$ with $F\left(  \psi\right)  <0.$ The inverse is also true, that is, to a minimally coupled scalar field it can be associated a unique non minimally coupled scalar field in a conformal metric and with a different potential function.*

***Proof:*** We first start with the action of Eq.(\[CL.12.0\]). Let $g_{ij}$ be the conformally related metric (this is not a coordinate transformation): $$g_{ij}=N^{-2}\bar{g}_{ij}.$$ Then the action provided by Eq.(\[CL.12.0\]) becomes [^2]:$$S_{NM}=\int d\tau dx^{3}N^{4}\sqrt{-g}\left[  F\left(  \psi\right)  \bar
{R}+\frac{\varepsilon}{2}N^{-2}g^{ij}\psi_{;i}\psi_{;j}-\bar{V}\left(
\psi\right)  \right]  .$$ Inserting the Ricci scalar $\bar{R}$ (using $n=4)$ from Eq.(\[CLN.04.1\]) into the latter equation we find:

$$\begin{aligned}
\label{ICLN}
S_{NM}
&=&\int d\tau dx^{3}N^{4}\sqrt{-g}\left[ F\left( \psi \right)
N^{-2}R-6F\left( \psi \right) N^{-3}\Delta _{2}N
+\frac{\varepsilon }{2}N^{-2}\Delta _{1}\psi -\bar{V}\left( \psi \right) \right]\,.
$$

Now we can define the conformal function $N$ in terms of the coupling function $F(\psi)$ \[where $F\left(  \psi\right)  <0)$\]: $$\label{CLN122}N=\frac{1}{\sqrt{-2F\left(  \psi\right)  }}~.$$ with $$\label{CLN123}N_{;i}=\frac{F_{\psi}\psi_{;i}}{\left(  -2F\right)  ^{\frac
{3}{2}}}\;.$$

Using Eqs.(\[CLN122\]) and (\[CLN123\]) the first term of the integral in Eq.(\[ICLN\]) becomes: $$\int 