  The two-dimensional extension of the one-dimensional PDM-Lagrangians and
their nonlocal point transformation mappings into constant unit-mass exactly
solvable Lagrangians is introduced. The conditions on the related
two-dimensional Euler-Lagrange equations' invariance are reported. The mappings
from superintegrable linear oscillators into sub-superintegrable nonlinear
PDM-oscillators are exemplified by, (i) a sub-superintegrable
Mathews-Lakshmanan type-I PDM-oscillator, for a PDM-particle moving in a
harmonic oscillator potential, (ii) a sub-superintegrable Mathews-Lakshmanan
type-II PDM-oscillator, for a PDM-particle moving in a constant potential, and
(iii) a sub-superintegrable shifted Mathews-Lakshmanan type-III PDM-oscillator,
for a PDM-particle moving in a shifted harmonic oscillator potential. Moreover,
the superintegrable shifted linear oscillators and the isotonic oscillators are
mapped into a sub-superintegrable PDM-nonlinear and a sub-superintegrable
PDM-isotonic oscillators, respectively.
