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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. Balińska, S.K. Simić, K.T. Zwierzyński: Which nonregular 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.The paper is dedicated to professor Dobrilo D -. Tošić on occasion of his 81st birthday.

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