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If $F(x,y)=0$ has infinitely many integral solutions, then by Siegel's Theorem the projective closure of this curve has at most $2$ points at infinity. These point are just those with coordinates $(x:y:0)$ with $H(x,y)=0$, where $H=\text{High}(F)$. So the polynomial $H$ has total degree at most $2$ if it is to be separable and not divisible by $x$ nor $y$.