For an elliptic curve $E$ over $\mathbb{Q}$, it is well-known that the torsion points on $E$ are integral points. 


Then, is it possible that there exists an example whose all of non-torsion rational points (or all of points of a subgroup of $E\_{free}(\mathbb{Q}$)) are integral points (of course, with respect to affine coordinate)? 


I never think that there is such an example and maybe it is a stupid question. 



An example of "all non-torsion rational points on an elliptic curve are integral points''?
It’s not possible for a much simpler reason than Siegel’s Theorem. Let $p$ be a prime of good reduction, and $\tilde X$ the reduction mod $p$ of your putative integral nontorsion point. Since $\tilde X$ is a torsion point mod $p$, say $[n](\tilde X)=\mathbb O$, the neutral point. Then $[n](X))$ is in the $p$-neighborhood of the neutral point, i.e. has $p$ in the denominator of its coordinates.


