  Within the standard Lagrangian settings (i.e., the difference between kinetic
and potential energies), we discuss and report isochronicity, linearizability
and exact solubility of some $n$-dimensional nonlinear position-dependent mass
(PDM) oscillators. In the process, negative the gradient of the PDM-potential
force field is shown to be no longer related to the time derivative of the
canonical momentum, $\mathbf{p}% =m\left( r\right) \mathbf{\dot{r}}$, but it is
rather related to the time derivative of the pseudo-momentum, $\mathbf{\pi
}\left( r\right) =\sqrt{% m\left( r\right) }\mathbf{\dot{r}}$ (i.e., Noether
momentum). Moreover, using some point transformation recipe, we show that the
linearizability of the $n$-dimensional nonlinear PDM-oscillators is only
possible for $n=1$ but not for $n\geq 2$. The Euler-Lagrange invariance falls
short/incomplete for $n\geq 2$ under PDM settings. Alternative invariances are
sought, therefore. Such invariances, like \emph{Newtonian invariance} of
Mustafa \cite{42}, effectively authorize the use of the exact solutions of one
system to find the solutions of the other. A sample of isochronous
$n$-dimensional nonlinear PDM-oscillators examples are reported.
