Oracle finding all integral points on genus 0 curves is a factoring oracle (e.g. $xy=n$ and $x^2-y^2=n$

I asked Can the number of solutions $xy(x−y−1)=n$ for x,y,n∈Z be unbounded as n varies? and occurred to me that an oracle giving all integral points may find nontrivial factor of $n$. Drama is this will not work for all $n$.

Would an oracle for finding all integral points on genus 1 curves (in whatever model) be:

- (loosely defined) Weak factoring oracle which finds at least one nontrivial factor
- Strong factoring oracle which finds all prime factors?

The factoring oracle must work for all integers if it exists.

(EDIT): Intuitively if I had genus 0 oracle for integral points I could factor general integers. If the oracle were for genus 1 I don't see a way for general integers but I would be lucky with integers of the form $xy(x-y-1)$ (just an example)