A Differential Integrability Condition for Two-Dimensional Hamiltonian Systems

We review, restate, and prove a result due to Kaushal and Korsch [Phys. Lett. A 276, 47 (2000)] on the complete integrability of two-dimensional Hamiltonian systems whose Hamiltonian satisfies a set of four linear second order partial differential equations. In particular, we show that a two-dimensional Hamiltonian system is completely integrable, if the Hamiltonian has the form $H=T+V$ where $V$ and $T$ are respectively harmonic functions of the generalized coordinates and the associated momenta.