In this paper, we complement the study of the stretch-twist-fold (STF) differential system, that depends on two nonnegative parameters, a >= 0 and b >= 0 . It models an STF flow, that consists of a family of Stokes flows in the unit ball arising in dynamo theory, and describes a mechanism for the generation and maintenance of magnetic fields in celestial bodies. The three-dimensional system of ordinary differential equations modelizing the STF flow is We study the global dynamics in the Poincar & eacute; ball of this differential system when a = 0 and b >= 0 , i.e. when the system is completely integrable, corresponding to the absence of quasi-rigid rotations. We determine all the local phase portraits of the equilibrium points, as well as the global phase portraits on the invariant unit sphere and on the two-dimensional invariant surfaces induced by its first integrals. Finally, we describe the dynamic on the infinite sphere when a >= 0 and b >= 0 , i.e. on the boundary of the Poincar & eacute; ball.
