may happen that one of them has to be taken as the physical one in a particular case and then such invariant quantity would not have a physical meaning. This situation often happen if cosmological solutions fit data and then related quantities are assumed as “physical” [@CapozzielloPrado]. On the other hand, if a scalar field describes an actual particle in a given frame, then its properties (e.g, its mass and couplings to other fields) would change in the conformally-related metric. This fact does not mean that its properties have no physical meaning. In other words, the identification of conformally invariant physical quantities is a very difficult task if it is not based on first principles.

With these considerations in mind, let us start with defining some geometrical structures that will be useful in the discussion.

A vector field $X^{a}$ is a Conformal Killing Vector [@Yano] (hereafter CKV) of the metric $g_{ij}$ if there exists a function $\psi\left(  x^{k}\right)  $ so that: $$\mathcal{L}_{X}g_{ij}=2\psi\left(  x^{k}\right)  g_{ij}\label{CLN.01}$$ where $\mathcal{L}_{X}$ is the Lie derivative. In case that $\psi_{;a}=0$ i.e. $\psi=$constant, the vector $X$ is called homothetic (hereafter HV) while if $\psi=0$ then the vector $X$ is a Killing vector (hereafter KV). In this context, 