This paper proposes a novel, integral-equation-based (IE-based) reconstruction method for magnetic resonance electrical properties tomography (MREPT). The proposed method can reconstruct the electric field and EPs simultaneously based on the Helmholtz decomposition of the electric field. In the proposed algorithm, we iteratively apply the projections onto the spaces in which the true electric field is contained. An advantage of the proposed method is that convergence is theoretically guaranteed in this iterative process. The efficacy of the proposed method is validated through numerical simulations.