Invariant algebraic surfaces of the Rikitake system

In this paper we use the method of characteristic curves for solving linear partial differential equations to study the invariant algebraic surfaces of the Rikitake system ẋ = -μx + y(z + β) ẏ = -μy + x (z - β) ż = α - xy. Our main results are the following. First, we show that the cofactor of any invariant algebraic surface is of the form rz + c, where r is an integer. Second, we characterize all invariant algebraic surfaces. Moreover, as a corollary we characterize all values of the parameters for which the Rikitake system has a rational or algebraic first integral.