= Solution
The separability of the printed <Lagrangian> is important. Every $x_i$ is an independent oscillator of the same <frequency>, irrespective of <mass>. Define $X_i=\sqrt{m_i}x_i$, $P_i=p_{x_i}/\sqrt{m_i}$. Then the complete constant matrix $Q_{ij}=P_iP_j+\omega^2X_iX_j$, and angular quantities $J_{ij}=X_iP_j-X_jP_i$, include each oscillator <energy> and the rotational symmetries of this mass-weighted oscillator space. Alternatively its explicitly time-dependent constants are $X_i\cos\omega t-P_i\sin\omega t/\omega$ and $X_i\sin\omega t+P_i\cos\omega t/\omega$.
<translation (geometry)>[Translations] in all $y_i$ give conserved $P_y=\sum p_{y_i}$. <translation (geometry)>[Translation] of $z_1,z_2$ together gives $P_{z12}=p_{z_1}+p_{z_2}$; $z_3$ is free and gives $p_{z_3}$. Their center-of-mass positions minus the associated constant velocities times $t$ are boost constants. The decoupled relative $z_1-z_2$ motion conserves $E_{z,\mathrm{rel}}=(p_{z_1}/m_1-p_{z_2}/m_2)^2\mu_{12}/2+V(z_1-z_2)$, where $\mu_{12}=m_1m_2/(m_1+m_2)$. In mass-weighted versions, the three free center coordinates also have conserved mutual <angular momenta>. These families are not all <functionally independent>.
The $y$ relative subsystem alone is driven through $\cos[\lambda(m_1-m_2)t]$. For generic nonzero $\lambda(m_1-m_2)$ its <energy>, and hence the total <energy>, is not conserved. Omitting the <mass-weighted oscillator symmetries> simply because the physical masses differ would miss genuine symmetries of this particular model.
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