Two-dimensional position-dependent mass Lagrangians; Superintegrability and exact solvability

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.