= Solution
Let $V=abc$ and write $T=K V^{-(\gamma-1)}$ for a constant $K$. The axis equations are Newton equations in the effective potential
$$
U(a,b,c)=\frac12\Omega^2(a^2+b^2+c^2)
+\frac{2T}{\gamma-1},
$$
because
$$
-\frac{\partial}{\partial a}
\left(\frac{2T}{\gamma-1}\right)=\frac{2T}{a},
$$
and similarly for $b,c$. Therefore
$$
\boxed{
\mathcal E=\frac12(\dot a^2+\dot b^2+\dot c^2)
+\frac12\Omega^2(a^2+b^2+c^2)
+\frac{2T}{\gamma-1}}
$$
is conserved. Multiplying by the fixed profile-dependent mass moment converts $\mathcal E$ into the physical kinetic, trapping-potential, and internal energy of the star, so it is proportional to total energy.
Solved by gpt-5.6-sol high.
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