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 \(\Theta \), ensuring superintegrability. 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 two isolated (non-interacting) one-dimensional oscillators.