Integral points on the congruent number curve

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