= Solution
Use the principal coordinates
$$
p=\frac{x_1+x_3}{\sqrt2},\qquad q=x_2,\qquad r=\frac{x_1-x_3}{\sqrt2}.
$$
Part (c) puts the equation in the canonical form
$$
\boxed{6p^2+3q^2+(2c-4)r^2=1.}
$$
If $c>2$, all three coefficients are positive: the surface is an <ellipsoid> with semiaxes $1/\sqrt6$, $1/\sqrt3$ and $1/\sqrt{2c-4}$ along the three <eigenvectors> listed in part (c).
If $c=2$, the equation is independent of $r$: it is an <elliptic cylinder> parallel to $(1,0,-1)$, with cross-sectional semiaxes $1/\sqrt6$ and $1/\sqrt3$.
If $c<2$, the form has two positive coefficients and one negative coefficient: it is a <one-sheet hyperboloid>, with axis $(1,0,-1)$ and waist ellipse $6p^2+3q^2=1$ at $r=0$. Its axial scale is $1/\sqrt{4-2c}$.
These are the three cases of the <signature transition from an ellipsoid to an elliptic cylinder and a one-sheet hyperboloid>. At the repeated <eigenvalues> $c=7/2$ and $c=5$, <principal axes> in the corresponding two-dimensional <eigenspace> are not unique: the <ellipsoid> is respectively an <oblate spheroid> about the $p$ axis or a <prolate spheroid> about the $q$ axis.
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