Integrability of a nuclear spin generator

In this work we consider a nuclear spin generator given byx˙=−βx+y,y˙=−x−βy(1−κs),s˙=β(α(1−s)−κy2), where α, β, κ are nonnegative parameters. It models the two temperature feedback nuclear reactor problem as model by Vreeke and Sandquist (1970) [4]. 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 β=0β=0 or β≠0β≠0 and κ=0κ=0 are the only values of the parameters for which the system is integrable, and in this case we provide an explicit expression for its first integrals.