On the Diophantine Equation $x_1^{3}-x_2^{2}x_1+1=0$ over $\mathbb{Q}(\sqrt{2})$

I investigate the Diophantine equation $x_1^{3}-x_2^{2}x_1+1=0$ where $x_1\in\mathbb{Q}(\sqrt{2})$ and $x_2\in\mathbb{Z}[\sqrt{2}]$. Using the arithmetic of the quadratic integer ring $\mathbb{Z}[\sqrt{2}]$, together with norm arguments, divisibility properties, and the explicit description of its unit group, I prove that the equation has exactly two solutions, namely $(x_1,x_2)=(-1,0)~\text{ and }~(1,\sqrt{2})$ As an application, I consider the family of elliptic curves $C_m:Y^{2}=X^{3}-m^{2}X+1,~ m\in\mathbb{Z}[\sqrt{2}],$ and deduce that, for every $m\neq0,\sqrt{2}$ the Mordell--Weil group $C_m(\mathbb{Q}(\sqrt{2}))$ contains no rational point of order two.