= Solution
For a <normal mode> write $x_j=A_j e^{i\omega t}$. The <generalized eigenvalue problem> is
$$
k\begin{pmatrix}1&-1&0\\-1&2&-1\\0&-1&1\end{pmatrix}A
=\omega^2m\operatorname{diag}(1,2,1)A.
$$
The three independent <eigenvectors> and corresponding squared frequencies are
$$
\boxed{(1,1,1):0;\qquad(1,0,-1):k/m;\qquad(1,-1,1):2k/m.}
$$
The last two have zero mass-weighted displacement and are <orthogonal> in the mass inner product. The lowest frequency is zero: it represents a rigid translation with both springs unchanged. Its general motion includes uniform centre-of-mass velocity, not just a constant displacement. Superpositions of the two oscillatory <normal modes> and this translation give the general solution.
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