= Solution
When $m_1=m_2$, the cosine becomes one and the previously driven $y$ relative subsystem becomes autonomous. Its <energy> is one new generic independent time-independent <constant of motion>[integral]; equivalently total <energy> becomes conserved because the remaining subsystem energies already were. Making $m_3$ equal as well introduces \b[no further generic <continuous symmetry>] in the printed model. The arbitrary $U_{13}$ and $V_{12}$ distinguish particle labels and directions, so equal masses alone do not create a permutation or physical-rotation invariance.
A finite “number of conserved quantities” must mean independent generators/<Constant of motion>[integrals]: arbitrary functions of any <constant of motion>[integral] give infinitely many. Mass-weighted rotations and free-coordinate symmetries identified in part a already exist at unequal masses. Thus the justified generic increment is one at $m_1=m_2$, and zero further at complete equality. Special potentials or $\lambda=0$ can change the count, but were not stipulated.
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