= Solution
Continue with $\mu=2$. Decompose the initial displacement into the <normal modes>:
$$
(2\varepsilon,-\varepsilon,0)^T=\varepsilon v_1+\varepsilon v_2.
$$
There is no translational component. Because all initial velocities vanish, the oscillatory coordinates contain only cosines and the zero-frequency velocity is zero. Therefore
$$
\boxed{\begin{aligned}
x_1(t)&=\varepsilon\big[\cos(\Omega t)+\cos(\sqrt2\,\Omega t)\big],\\
x_2(t)&=-\varepsilon\cos(\sqrt2\,\Omega t),\\
x_3(t)&=\varepsilon\big[-\cos(\Omega t)+\cos(\sqrt2\,\Omega t)\big],\qquad \Omega=\sqrt{k/m}.
\end{aligned}}
$$
At $t=0$ these give the prescribed displacements and zero velocities. Each term separately satisfies $M\ddot x+Kx=0$. Also $x_1+2x_2+x_3=0$ for all $t$, so the <centre of mass> remains at its original equilibrium position, consistent with the absence of external forces.
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