Integrability of the Nose-Hoover equation

In this work we consider the Nose-Hoover equation for a one dimensional oscillator (x) over dot = -y - xz, (y) over dot = x, (z) over dot = alpha(x(2) - 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 alpha = 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. (C) 2011 Elsevier B.V. All rights reserved.