= Solution
Write the displacement as $\mathbf x=\mathbf a\cos(\omega t+\phi)$. The <generalized eigenvalue problem> is
$$
k\begin{pmatrix}1&-1&0\\-1&2&-1\\0&-1&1\end{pmatrix}\mathbf a=\omega^2\operatorname{diag}(m,M,m)\mathbf a.
$$
The symmetry of the two end atoms suggests an end-antisymmetric shape $(1,0,-1)$ and two end-symmetric shapes. Direct substitution gives the <normal modes of three masses connected by two equal springs>:
$$
\boxed{\begin{array}{c|c}
\mathbf a&\omega^2\\\hline
(1,1,1)&0\\
(1,0,-1)&k/m\\
(1,-2m/M,1)&k/m+2k/M
\end{array}}
$$
These three independent <eigenvectors> exhaust the three-dimensional system. The ordinary <frequencies> are $\omega/(2\pi)$; the values $\omega$ above are <angular frequencies>.
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