An alternative approach has been developed to the numerical synthesis of ion-beam focusing systems that prepare the dense laminar ion beams with the required profile of the ion trajectories. Conventionally an ill-posed synthesis problem arises in such cases, and only low-density beams with no magnetic field are treated. Instead, we compute both the electric and magnetic fields by considering two well-posed problems, the first for the equivalent potential Q and the second for the electric potential U. An analytical solution for Q for the specific case of an axial system has been found and a self-consistent method of satisfying the axial boundary conditions for Poisson's equation in U is described. It is shown that the beam can be focused while preserving its laminar structure and the required profile when applying various superpositions of both the electric and magnetic fields.