This note studies Hamiltonian systems which are thermostated using the Jellinek—Berry thermostat (J. Chem. Phys. 1988; Phys. Rev. A 1988). Jellinek & Jellinek and Berry propose an extension of Nosé's thermostat (J. Chem. Phys. 1984). They introduce multiple functional parameters in order to achieve ergodicity of the thermostatted dynamics. This family of Hamiltonian thermostats aim to simulate the macro canonical ensemble of a Hamiltonian $H$ by coupling $H$ to a 1-d heat reservoir with potentialenergy $v(s)$ and kinetic energy $p^2/2Q(s)$. Our note derives a normal form for the reservoir’s potential energy; investigates when the Jellinek—Berry thermostated system admits a Nosé—Hoover reduction; and, we demonstrate that a Jellinek—Berry thermostated periodic ideal gas is completely integrable and satisfies a KAM twist condition called Rüssmann non-degeneracy. This is used to deduce that a thermostated, collision-less, non-ideal gas (i.e. one with a smooth potential energy) at sufficiently high temperatures of the reservoir has a positive measure set of invariant tori—hence, the thermostated dynamics are non-ergodic.