A geometric formulation is adopted for a nonlinear magnetohydrodynamic system wherein the magnetic field is aligned with the direction of the binormal to the streamlines. It is established that, for complex-lamellar motion, if the divergence of the binormal field vanishes then the fluid streamlines are geodesics on generalized helicoids. The latter constitute the Maxwellian surfaces and the magnetic lines are helices thereon. The key geometric and physical parameters of the magneto-hydrodynamic motion are all determined in terms of the torsion τ of the streamlines. A superposition principle is presented whereby a more general class of magnetohydrodynamic motions may be isolated with streamlines and magnetic lines no longer restricted to be geodesics or parallels on the Maxwellian surfaces. Moreover, the class so generated is not subject to the complex-lamellar constraint. In an appendix, a particular reduction is obtained to the integrable Da Rios system.