Perhaps the situation becomes clearer by looking at a similar problem. Replacing the circle by a hyperbola, we can look at rational points on $H: XY = R$. This conic has the obvious point $N = (1,R)$, and among the many rational points on $H$ there are a few integral points, each corresponding to a factorization of $R$. You can define a group law on $H$ with neutral element, but this does not help at all at finding "nontrivial" rational points (that is, integral points). Neither does it seem to help to replace the rationals by an algebraic number field. In fact the geometric picture does not seem to add to our understanding in this case (on this elementary level, at least).
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