}.\label{CLN.08}$$

Now if we consider a conformal transformation of the metric $\bar{g}_{ij}=N^{2}\left(  x^{k}\right)  g_{ij}$ and a new potential function $\bar
{V}\left(  x^{k}\right)  =\frac{V\left(  x^{k}\right)  }{N^{2}\left(
x^{k}\right)  }$ then the new Lagrangian $\bar{L}\left(  x^{k},x^{\prime
k}\right)  $ takes the following form: $$\bar{L}\left(  x^{k},x^{\prime k}\right)  =\frac{1}{2}\bar{g}_{ij}x^{\prime
i}x^{\prime j}-\bar{V}\left(  x^{k}\right)  \label{CLN.09}$$ implying that Eq.(\[CLN.09\]) is of the same form as the Lagrangian $L$ in Eq.(\[CLN.05\]). From now on *the Lagrangian $L\left(  x^{k},\dot
{x}^{k}\right)  ~$ of Eq.(\[CLN.05\]) and the Lagrangian $\bar{L}\left(
x^{k},x^{\prime k}\right)  $ of Eq.(\[CLN.09\]) will be called conformal*. In this framework, the action remains the same i.e. it is invariant under the change of parameter, the equations of motion in the new variables $(\tau
,x^{i})$ will be the same with the equations of motion for the Lagrangian $L$ in the original coordinates $(t,x^{k})$ .

It has been shown [@TsamGE] that the Noether symmetries of a Lagrangian of the form (\[CLN.05\]) follow the homothetic algebra of the metric $g_{ij}.$ The same applies to the Lagrangian $\bar{L}\left(  x^{k},x^{\prime k}\right)
$ and the metric $\bar{g}_{ij}.$ As we have remarked the conformal algebra of the metrics $g_{ij},\bar{g}_{ij}$ (as a set) is the same however their closed subgroups of HVs and KVs are in general different[^1]. Now, we 