 Solution

Substituting $\omega =U{\omega }^{\prime }$ and using the diagonalization givesFor $I_0>0$ and energy level one in the chosen units, its ellipse equation isThe major semiaxis has length $a=\sqrt{2/I_0}$ along $u_1=(1,1)/\sqrt{2}$; the minor semiaxis has length $b=\sqrt{2/(3I_0)}$ along $u_2=(-1,1)/\sqrt{2}$. The full axis lengths are $2a$ and $2b$. The original-coordinate curve is the same ellipse tilted by $45^{\circ }$, elongated along ${\omega }_y={\omega }_x$; an orthogonal coordinate change preserves lengths and does not change the ellipse itself.