= Solution
For weak anchoring, $k_2\ll k_1$, use the normalized free-molecule <eigenvectors> $u_T=(1,1)/\sqrt2$ and $u_S=(1,-1)/\sqrt2$. The perturbation is
$$
\delta D=\frac{k_2}{m}\begin{pmatrix}1&0\\0&0\end{pmatrix}.
$$
Its diagonal <matrix> elements in this basis are both $k_2/(2m)$. First-order <time-independent perturbation theory> for the <eigenvalues> therefore gives
$$
\boxed{\omega_T^2\simeq\frac{k_2}{2m},\qquad
\omega_S^2\simeq\frac{2k_1}{m}+\frac{k_2}{2m}.}
$$
Hence \b[$\omega_T\simeq\sqrt{k_2/(2m)}$] and \b[$\omega_S\simeq\sqrt{2k_1/m}\,[1+k_2/(8k_1)]$]. The zero free translation requires taking a square root of its new <eigenvalue>, rather than Taylor-expanding the <frequency> about zero. It is not a degenerate <eigenvalue> in the stipulated one-dimensional two-coordinate model.
As a check, the exact squared <frequencies> are
$$
\omega_\pm^2=\frac{2k_1+k_2\pm\sqrt{4k_1^2+k_2^2}}{2m}.
$$
Expansion gives the same first-order results; the next lower-mode correction is $-k_2^2/(8k_1m)$. The perturbed <eigenvectors> contain small translation/stretch admixtures of order $k_2/(4k_1)$, consistent with the failure of uniform translation to remain an exact mode in orientation (1).
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