= Solution
The <matrix> is $M=2I-ee^T$, where $e=(1,1,1)^T$. Along $\hat e=e/\sqrt3$, its <eigenvalue> is $2-3=-1$. Every <vector> perpendicular to $e$ has <eigenvalue> two. For example, use the orthonormal basis
$$
u_1=\frac{(1,-1,0)}{\sqrt2},\qquad u_2=\frac{(1,1,-2)}{\sqrt6},\qquad u_3=\frac{(1,1,1)}{\sqrt3}.
$$
The principal-coordinate equation is
$$
2\xi^2+2\eta^2-\zeta^2=3,\qquad\frac{\xi^2+\eta^2}{3/2}-\frac{\zeta^2}{3}=1.
$$
It depends only on distance from the $u_3$ axis and position along it, so every rotation about that axis preserves the surface. Therefore
$$
\boxed{\text{the rotational axis is the line }x_1=x_2=x_3,\text{ with direction }(1,1,1).}
$$
This is a <one-sheet hyperboloid> with waist radius $\sqrt{3/2}$. The repeated positive <eigenvalue> makes the two transverse <principal axes> nonunique.
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