Integrability of the Hide–Skeldon–Acheson dynamo

In this work we consider the Hide–Skeldon–Acheson dynamo modelx˙=x(y−1)−βz,y˙=α(1−x2)−κy,z˙=x−λz, where α, β, κ and λ   are parameters. We contribute to the understanding of its global dynamics, or more precisely, to the topological structure of its orbits by studying the integrability problem. Provided α≠0α≠0 we identify the values of the parameters of this model, for which it admits a first integral. Also, as a corollary of our main results we get that for α,β,κ≠0α,β,κ≠0 the dynamo model does not admit a polynomial, rational or Darboux first integral.