A study of the averaged magnetic-surface equations leads to analytical expressions for the conditions of absence of an island structure of the vacuum magnetic surfaces in the paraxial region of an ℓ = 3 torsatron magnetic field. An analysis of these expressions allows the choice of the optimum modulation for the angle of the helical conductor winding, which considerably reduces the resulting transverse magnetic field required to satisfy the condition for no or minimal island structure. In this case, the displacement of the torsatron magnetic axis from the circular torus axis is much smaller than in the absence of modulation.