= Solution
With $m_1=m_2=m_3=m$, the explicit <energy> and ordinary <rotation> generators just discussed are
$$
\boxed{\begin{aligned}
H={}&\sum_i\frac{p_{x_i}^2+p_{y_i}^2+p_{z_i}^2}{2m}
+\frac{m\omega^2}{2}\sum_i x_i^2\\
&+U_{12}(y_1-y_2)+U_{13}(y_1-y_3)+V(z_1-z_2),\\
J_{12}&=x_1p_{x_2}-x_2p_{x_1},\\
J_{23}&=x_2p_{x_3}-x_3p_{x_2},\qquad
J_{31}=x_3p_{x_1}-x_1p_{x_3}.
\end{aligned}}
$$
The <energy> is constant because the drive factor no longer depends on time. For each <rotation> charge, differentiating and using $\ddot x_i=-\omega^2x_i$ gives $\dot J_{ij}=m(x_i\ddot x_j-x_j\ddot x_i)=0$.
These are angular <momenta> in the auxiliary three-dimensional space whose coordinates are the particle labels $(x_1,x_2,x_3)$. They are not the physical spatial <angular momentum> $\sum_i\mathbf r_i\times\mathbf p_i$; generic $U_{12},U_{13},V$ do not have physical rotational invariance. They are precisely the equal-mass representatives of the weighted charges already present in <2/a/solution>. The new <energy> integral supplies the autonomous relative-$y$ dynamics not present in the driven case.
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