Determine all pairs of integers $(a, b)$ such that \[\frac{a^{2}}{2ab^{2}-b^{3}+1}\] is a positive integer.
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Tags: Divisibility Theory
Determine all pairs of integers $(a, b)$ such that \[\frac{a^{2}}{2ab^{2}-b^{3}+1}\] is a positive integer.