Generalized Pell solutions fall into finitely many Pell orbits

Put $t:=\frac{\mathrm{\mid }\alpha \mathrm{\mid }}{\sqrt{\mathrm{\mid }N\mathrm{\mid }}}>0.$ Because ${\epsilon }_D>1$, the half-open intervals $({\epsilon }_D^{m-1\mathrm{/}2},{\epsilon }_D^{m+1\mathrm{/}2}](m\in \mathbb{Z})$ partition $(0,\mathrm{\infty })$: their endpoints are strictly ordered, they tend to $0$ as $m\to -\mathrm{\infty }$, and they tend to $+\mathrm{\infty }$ as $m\to +\mathrm{\infty }$. Hence there is a unique integer $k$ with ${\epsilon }_D^{-1\mathrm{/}2}<t{\epsilon }_D^{-k}\le {\epsilon }_D^{1\mathrm{/}2}\mathrm{.}$ Put ${\alpha }_0:=\alpha {\epsilon }_D^{-k}=x_0+y_0\sqrt{D}\mathrm{.}$ By [F1] and [F2], ${\alpha }_0$ is Pell-equivalent to $\alpha$ and still satisfies $N_D({\alpha }_0)=N\mathrm{.}$

Let $\beta :=\mathrm{\mid }{\alpha }_0\mathrm{\mid }\mathrm{.}$ Step 1.1 gives $\sqrt{\mathrm{\mid }N\mathrm{\mid }}{\epsilon }_D^{-1\mathrm{/}2}<\beta \le \sqrt{\mathrm{\mid }N\mathrm{\mid }}{\epsilon }_D^{1\mathrm{/}2}\mathrm{.}$ Since $\overline{{\alpha }_0}=x_0-y_0\sqrt{D}=\frac{N}{{\alpha }_0},$ one has $\mathrm{\mid }\overline{{\alpha }_0}\mathrm{\mid }=\frac{\mathrm{\mid }N\mathrm{\mid }}{\beta }\le \sqrt{\mathrm{\mid }N\mathrm{\mid }}{\epsilon }_D^{1\mathrm{/}2}$ and also $\mathrm{\mid }\overline{{\alpha }_0}\mathrm{\mid }>\sqrt{\mathrm{\mid }N\mathrm{\mid }}{\epsilon }_D^{-1\mathrm{/}2}$. Therefore $2\mathrm{\mid }{x}_0\mathrm{\mid }=\mathrm{\mid }{\alpha }_0+\overline{{\alpha }_0}\mathrm{\mid }\le \beta +\mathrm{\mid }\overline{{\alpha }_0}\mathrm{\mid }\le \sqrt{\mathrm{\mid }N\mathrm{\mid }}(\sqrt{{\epsilon }_D}+{\epsilon }_D^{-1\mathrm{/}2}),$ and similarly $2\sqrt{D}\mathrm{\mid }{y}_0\mathrm{\mid }=\mathrm{\mid }{\alpha }_0-\overline{{\alpha }_0}\mathrm{\mid }\le \beta +\mathrm{\mid }\overline{{\alpha }_0}\mathrm{\mid }\le \sqrt{\mathrm{\mid }N\mathrm{\mid }}(\sqrt{{\epsilon }_D}+{\epsilon }_D^{-1\mathrm{/}2})\mathrm{.}$ This is exactly the stated bound.

The bounds of step 2.1 leave only finitely many integer pairs $(x_0,y_0)$. Every solution is Pell-equivalent to one of them by step 1.1, so only finitely many Pell-equivalence classes occur.