Three. (20 points) Let the line $y=x+\sqrt{2}$ intersect the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ at points $M$ and $N$, and $O M \perp O N$ ($O$ is the origin). If $M N=\sqrt{6}$, find the equation of the ellipse.