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An integral formulation for eddy-current problems in nomagnetic structures is presented. The solenoidality of the current density is assured by introducing a current vector potential T. This potential possesses only two scalar components, as the gauge chosen to ensure its uniqueness is T. u = 0, where u is a prescribed vector field. The discrete analogue of this gauge and the boundary conditions are directly imposed by the shape functions, exploiting the use of edge finite elements and the methods of network theory. In the frame of the integral methods, this approach seems the most adequate to analyse the eddy currents induced in both massive conductors and thin shells. In massive structures, the two degrees of freedom are to be compared to four of the usual integral methods which exploit the presence of a scalar potential to ensure solenoidality. On the other hand, the procedure naturally reduces to the stream function approach when applied to thin shells. Finally, an integration procedure which guarantees symmetry and positive-definiteness of the inductance matrix is proposed.