= Solution
At $m_1=m_2$, the explicitly time-dependent coefficient becomes one. This restores <time-translation symmetry>, and the total <energy> becomes conserved. No further analogous restoration occurs merely from setting $m_3$ equal to them: the three arbitrary interaction functions still distinguish their arguments and particle labels.
If the question counts the extra \b[ordinary unweighted <rotations>] of the $x$ coordinate labels, the conventional generator count is as follows. For $m_1=m_2\ne m_3$, <rotations> of $(x_1,x_2)$ preserve both the kinetic and oscillator terms, so one ordinarily displayed <rotation> charge joins the restored <energy>: two displayed generators beyond the elementary list. For three equal masses, the full ordinary <rotation> group of $(x_1,x_2,x_3)$ has three generators; together with <energy> this gives four, or two more than in the preceding equal-pair case.
There is a necessary qualification to interpreting those numbers as genuinely new independent conservation laws. The weighted charges in <2/a/solution> deform continuously to those same ordinary <rotation> charges at equality. For example $\mathcal J_{12}$ is conserved for every positive $m_1,m_2$ and becomes $x_1p_{x_2}-x_2p_{x_1}$ when the masses agree. Thus
$$
\boxed{\begin{gathered}
\text{Restricted ordinary-coordinate generator count: }2\text{ and }4.\\
\text{New generic energy conservation: }1\text{ at }m_1=m_2;\quad
0\text{ further at }m_3=m_1.
\end{gathered}}
$$
The restriction must be stated if using the first count. Equal <mass> does not create three previously absent independent oscillator integrals. If $\lambda=0$, <energy> conservation was already present; special choices of the interaction functions can add further symmetries and are excluded by the generic assumption.
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