  We present the analysis of one week of single-site high-speed CCD photometric
observations of the cool ZZ Ceti star HS 0507+0434B. Ten independent
frequencies are detected in the star's light variations: one singlet and three
nearly-equally spaced triplets. We argue that these triplets are due to
rotationally split modes of spherical degree l=1. This is the first detection
of consistent multiplet structure in the amplitude spectrum of a cool ZZ Ceti
star and it allows us to determine the star's rotation period: 1.70 +/- 0.11 d.
  We report exactly equal frequency, not period, spacings between the detected
mode groups. In addition, certain pairs of modes from the four principal groups
have frequency ratios which are very close to 3:4 or 4:5; while these ratios
are nearly exact (within one part in 10^4), they still lie outside the computed
error bars. We speculate that these relationships between different frequencies
could be caused by resonances. One of the three triplets may not be constant in
amplitude and/or frequency.
  We compare our frequency solution for the combination frequencies (of which
we detected 38) to Wu's (1998, 2001) model thereof. We obtain consistent
results when trying to infer the star's convective thermal time and the
inclination angle of its rotational axis. Theoretical combination-frequency
amplitude spectra also resemble those of the observations well, and direct
theoretical predictions of the observed second-order light-curve distortions
were also reasonably successful assuming the three triplets are due to l=1
modes. Attempts to reproduce the observed combination frequencies adopting all
possible l=2 identifications for the triplets did not provide similarly
consistent results, supporting their identification with l=1.
