If you multiply this by $2$ you get $(2x)^2 - 2N y^2 = -2$, so you can solve $x^2 - 2N y^2 = -2$ and then check when $x$ is even. $x^2 - 2Ny^2 = -2$ is a generalized Pell's equation, so you can find a ...
One more way to solve the problem. Let $x = 4t + 3$. Then
$$x^3 - 2 = 16t^2(4t + 9) + (108t + 25).$$
$$4t + 9 = a^2 \qquad 108t + 25 = b^2$$
has infinitely many solutions. It is reduced to ...
It has been known since Euler that the quartic surface defined by
$\displaystyle x_1^4 + x_2^4 = x_3^4 + x_4^4$
contains a rational curve defined by
$\displaystyle x_1(t) = t^7 + t^5 - 2t^3 + 3t^2 + t,...