Integral points on elliptic curves $y^{2}=x(x-2^{m}) (x+p)$

We provide a description of the integral points on elliptic curves $y^{2}=x(x- 2^{m}) \times (x+p)$, where $p$ and $p+2^{m}$ are primes. In particular, we show that for $m=2$ such a curve has no nontorsion integral point, and for $m=1$ it has at most one such point (with $y \gt 0$). Our proofs rely upon numerical computations and a variety of results on quartic and other diophantine equations, combined with an elementary analysis.