As the title says. Can we determine all the integral points on elliptic curves of the form
$$y^2=x^3+px$$
for a prime $p$? If yes, can someone explain me how? A good reference would also be sufficient.
This is completely worked out in Walsh's paper:
In particular, for your elliptic curve $y^2=x^3+px$, there are at most 2 primitive integer solutions if $p>3$, and 4 if $p=3$.
As for imprimitive solutions in this case ($k=1$, $d=p$), the proof of theorem 4 indicates that such a solution would be a counterexample to the Ankeny-Artin-Chowla conjecture.