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A graph is integral if the spectrum (of its adjacency matrix) consists  entirely of integers. The problem of determining all non-regular bipartite  integral graphs with maximum degree four which do not have ±1 as eigenvalues  was posed in K.T. Balinska, S.K. Simic, K.T. Zwierzynski: Which  non-regular bipartite integral graphs with maximum degree four do not have  ±1 as eigenvalues?, Discrete Math., 286 (2004), 15{25. Here we revisit this  problem, and provide its complete solution using mostly the theoretical  arguments.

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