Practical solution of the Diophantine equation $y^2 = x(x+2^ap^b)(x-2^ap^b)$

Let $p$ be an odd prime and $a$, $b$ positive integers. In this note we prove that the problem of the determination of the integer solutions to the equation $y^2 = x(x+2^ap^b)(x-2^ap^b)$ can be easily reduced to the resolution of the unit equation $u+\sqrt{2}v = 1$ over $\mathbb{Q}(\sqrt{2},\sqrt{p})$. The solutions of the latter equation are given by Wildanger's algorithm.