= Solution
A spectral component shifts by the difference between its upper- and lower-level Stark shifts. With $\Delta J=+1$, $\Delta K=0$ and $\Delta M=q$, the supplied energy formula gives
$$
\delta\widetilde\nu=\frac{\mu E_{\rm field}}{hc}K\left[\frac{M+q}{(J+1)(J+2)}-\frac{M}{J(J+1)}\right].
$$
For radiation polarized parallel to the static field, $q=0$, so
$$
\boxed{\delta\widetilde\nu=-\frac{2\mu E_{\rm field}KM}{hc\,J(J+1)(J+2)}.}
$$
Thus \b[the assigned $K=0$ feature has no first-order splitting], whereas the $K\ne0$ features acquire symmetrically arranged magnetic-sublevel components with spacings proportional to $|K|$. With perpendicular radiation, $q=\pm1$ gives the corresponding different allowed pattern. Recording field-dependent spectra with specified <polarization> would test both the J values and the K sequence; using an individual level's shift alone would not predict the observed line positions. Higher-order shifts of the nominally unsplit feature are outside this first-order test.
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