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The problem of determining the achievable rate region for an arbitrary source network with one "helper" is still unsolved. Csiszár and the author have shown that it reduces to the one-parameter entropy characterization problem (OPEC), treated in their monograph on information theory. For a discrete memoryless multiple source, solving the OPEC problem means finding a computable characterization of the per-letter conditional entropies of the first n outputs of each of the component sources given an arbitrary function of the first n outputs of the first component source. For sources with three components, the OPEC problem has been solved by Csiszár, Körner, and Marton. However, their result has a very asymmetric form and has not been generalized. This paper gives a substantially simpler proof of the same result in a new symmetric form. Moreover, for sources with more than three components, a new increased region of simultaneously attainable conditional entropies is derived.