The elliptic curve $-4 x^3 + 4 x^2 y + 16 x - y^3 + 9 y$ goes through $21$ integer points in the range $-9$ to $9$. Is that the maximum?
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This is a bit of a longshot, but have a look at Matthew Baker and Clayton Petsche, Global discrepancy and small points on elliptic curves, http://arxiv.org/pdf/math/0507228v1 and at some of the papers in the references that have titles that suggest they may be relevant to this problem.