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A novel integral equation satisfied by a single unknown surface current density is formulated for the two-dimensional analysis of the quasi-stationary field in systems of parallel conductors. This is an alternative to the coupled boundary integral equations formulated in terms of two unknowns, the magnetic vector potential and its normal derivative over each conductor surface. The accuracy of the results computed by the proposed solution method is demonstrated by comparison with results from exact analytical solutions. In order to demonstrate the increased efficiency of the proposed method, various test results are also compared with those obtained from the existent boundary integral equations. Significant reductions in the computation time have been achieved.