Integrability of the Nosé–Hoover equation

In this work we consider the Nosé–Hoover equation for a one dimensional oscillator ẋ=−y−xz,ẏ=x,ż=α(x2−1).It models the interaction of a particle with a heat-bath. We contribute to the understanding of its global dynamics, or more precisely, to the topological structure of its orbits by studying the integrability problem. We prove that α=0α=0 is the only value of the parameter for which the system is integrable, and in this case we provide an explicit expression for its first integrals.