Jacobi last multiplier and two-dimensional superintegrable oscillators

In this paper, we examine the role of the Jacobi last multiplier in the context of two-dimensional oscillators. We first consider two-dimensional unit-mass oscillators admitting a separable Hamiltonian description, i.e., $H=H_1+H_2$, where $H_1$ and $H_2$ are the Hamiltonians of two one-dimensional unit-mass oscillators; it is shown that there exists a third functionally-independent first integral $\mathrm{\Theta }$, thereby ensuring superintegrablility. Various examples are explicitly worked out. We then consider position-dependent-mass oscillators and the Bateman pair, where the latter consists of a pair of dissipative linear oscillators. Quite remarkably, the Bateman pair is found to be superintegrable, despite admitting a Hamiltonian which cannot be separated into those of two isolated (non-interacting) one-dimensional oscillators.