= Solution
Let the equilibrium radius be $a_0=b_0=c_0$ and $2T_0=\Omega^2a_0^2$. Linearizing the $a$ equation gives
$$
\boxed{
\frac{\delta\ddot a}{a_0}
=\Omega^2\left(-2\frac{\delta a}{a_0}
+\frac{\delta T}{T_0}\right)},
$$
with cyclic analogues. The temperature scaling gives
$$
\frac{\delta T}{T_0}
=-(\gamma-1)
\left(\frac{\delta a}{a_0}
+\frac{\delta b}{a_0}
+\frac{\delta c}{a_0}\right).
$$
For the <affine breathing mode of a star>, all three fractional axis changes equal $q$. Then
$$
\ddot q+(3\gamma-1)\Omega^2q=0,
\qquad
\boxed{\omega=\Omega\sqrt{3\gamma-1}}.
$$
This is the homologous compressional mode, which changes volume, density, and temperature.
For either independent <affine quadrupole mode of a star>, the three fractional changes sum to zero. Then $\delta T=0$ and
$$
\ddot q+2\Omega^2q=0,
\qquad
\boxed{\omega=\sqrt2\,\Omega}.
$$
These two degenerate modes deform the sphere into an ellipsoid while preserving its volume to first order.
Solved by gpt-5.6-sol high.
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