= Solution
Assume positive, pairwise distinct masses and generic nonzero $\omega,\lambda$. Let $p_{x_i}=m_i\dot x_i$, and similarly for $y,z$. The three uncoupled $x$ oscillators each conserve
$$
\boxed{E_{x_i}=\frac{p_{x_i}^2}{2m_i}+\frac12m_i\omega^2x_i^2\qquad(i=1,2,3).}
$$
Uniform translation of all $y_i$ conserves $P_y=\sum_i p_{y_i}$. The $z_1,z_2$ interaction permits their common translation, while $z_3$ is entirely free, giving two independent <momenta> $P_{z,12}=p_{z_1}+p_{z_2}$ and $p_{z_3}$. The interacting $z_1,z_2$ subsystem is autonomous and has its own conserved <energy>. A useful basic collection is therefore
$$
\boxed{P_y,\quad P_{z,12},\quad p_{z_3},\quad
E_{z,12}=\frac{p_{z_1}^2}{2m_1}+\frac{p_{z_2}^2}{2m_2}+V(z_1-z_2),\quad E_{x_1},E_{x_2},E_{x_3}.}
$$
The total <energy> is generally not conserved. With $\kappa=\lambda(m_1-m_2)$, its derivative is $\dot H=-\kappa\sin(\kappa t)U_{12}(y_1-y_2)$, which is generically nonzero.
\b[Unequal masses do not remove every <rotation> charge in the oscillator sector.] Put $X_i=\sqrt{m_i}x_i$, $P_i=p_{x_i}/\sqrt{m_i}$. This is a <canonical transformation> and makes that sector an <isotropic harmonic oscillator>. Its <rotations> conserve
$$
\boxed{\mathcal J_{ij}=X_iP_j-X_jP_i
=\sqrt{\frac{m_i}{m_j}}x_ip_{x_j}-\sqrt{\frac{m_j}{m_i}}x_jp_{x_i}.}
$$
These <mass-weighted oscillator symmetries> exist already at distinct masses. The energies and three <rotation> charges are not all functionally independent; the three-oscillator sector has at most five independent time-independent integrals on regular <trajectories>.
There are further simple explicitly time-dependent constants, such as $x_i\cos\omega t-(\dot x_i/\omega)\sin\omega t$ and $x_i\sin\omega t+(\dot x_i/\omega)\cos\omega t$, and center-of-mass boost constants. For example $MY-tP_y$ is constant for $M=\sum_im_i$ and $Y=\sum_im_iy_i/M$. These illustrate why an unrestricted raw number of conserved functions is not meaningful.
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