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Integrability of the Hide--Skeldon--Acheson dynamo

In this work we consider the Hide-Skeldon-Acheson dynamo model \[ \dot x=x(y-1)-βz, \quad \dot y =α(1-x^2)-κy, \quad \dot z =x-λz, \] 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 $α\ne 0$ we identify the values of the parameters of this model, for which it admits a first integral. Also, as corollary of our main results we get that for $α, β, κ\ne 0$ the dynamo model does not admit a polynomial, rational or Darboux first integral.