= Solution
For equal masses $m$, the newly conserved relative $y$ <energy> is
$$
\boxed{E_{y,\mathrm{rel}}=\sum_i\frac{p_{y_i}^2}{2m}-\frac{(\sum_i p_{y_i})^2}{6m}+U_{12}(y_1-y_2)+U_{13}(y_1-y_3).}
$$
It is <kinetic energy> relative to the free $y$ <center of mass> plus the two interactions. Adding its constant center <kinetic energy> and the already conserved $x,z$ energies gives the total <Hamiltonian>. The equal-mass oscillator rotations simplify to $J_{ij}=x_i p_{x_j}-x_jp_{x_i}$, but are not new: for unequal masses their conserved predecessors contain the mass-weight factors. As a concrete check, independent transformations of $X_i=\sqrt{m_i}x_i$ rotate an isotropic oscillator at \i[any] positive masses. Complete <mass> equality does not magically make generic $U_{12}(y_1-y_2)+U_{13}(y_1-y_3)+V(z_1-z_2)$ invariant under rotations of physical space.
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