On the dynamics of the Craik-Okamoto and the Euler top

We study the dynamics of the Craik-Okamoto system and its relation with the dynamics of the Euler top. We show that both systems exhibit the same dynamics in a neighborhood of infinity and we describe completely the phase portraits of the Euler top. Additionally we provide explicitly the Euler top solutions in function of the time. We show that the orbits given by the invariant straight lines of the Craik-Okamoto system are in fact center manifolds of equilibrium points at infinity. Moreover, we show that while in the Euler top all the orbits lie on invariant algebraic surfaces, in the Craik-Okamoto system any orbit is on an invariant algebraic surface.