Integral points on the congruent number curve

We study integral points on the quadratic twists $\mathcal{E}_D:y^2=x^3-D^2x$ of congruent number curve. give upper bounds in each coset $2\mathcal{E}_D(\mathbb{Q})$ $\mathcal{E}_D(\mathbb{Q})$ and show that their total is $\ll (3.8)^{\mathrm{rank} \mathcal{E}_D(\mathbb{Q})}$. further average non-torsion this family bounded above by $2$. As an application we also deduce from our system simultaneous Pell equations $aX^2-bY^2=d$, $bY^2-cZ^2=d$ for pairwise coprime positive integers $a,b,c,d$, has at most (3.6)^{\omega(abcd)}$ integer solutions.