The problem of plasma current density and safety factor reconstruction using magnetic field measurements is considered. In the traditional formulation, the problem is strongly ill-posed. In particular, substantially different current densities and safety factors can be equally well attributed to the same set of measurements. The paper presents an accurate mathematical formulation of the inverse problem and its variants. A numerical algorithm is given, which allows to find all substantially different solutions or to prove the absence of multiple solutions. Examples of very different current density and safety factor reconstructions for measurements with finite accuracy are presented. Cases of MAST, JET and ITER-like plasmas are considered. It is shown that including the Motional Stark Effect (MSE) measurements as constraints, provided the accuracy of MSE measurements is sufficient, allows identifying one solution. The approach of the paper can be used for a wide range of inverse problems in physics and help in selecting additional conditions, which can identify the most likely solution among several.