 Solution

An orthogonal transformation to the eigenvectors preserves distance from the centre and writes the surface as $u^2+v^2+3w^2=3$. It is an ellipsoid with semi-axes $\sqrt{3},\sqrt{3},1$. ThereforeThe original $y$ axis is one of the eigen-directions with eigenvalue one, and the centre has $a_y=0$. Consequently $y^2\le 3$, attained when the other centred coordinates vanish, so $\boxed{y_{\max }=\sqrt{3}}$.