  The subdwarf-B pulsator, KIC 10553698A, is one of 16 such objects observed
with one-minute sampling for most of the duration of the Kepler Mission. Like
most of these stars, it displays a rich g-mode pulsation spectrum with several
clear multiplets that maintain regular frequency splitting. We identify these
pulsation modes as components of rotationally split multiplets in a star
rotating with a period of about 41 d. From 162 clearly significant
periodicities, we are able to identify 156 as likely components of l=1 or l=2
multiplets. For the first time we are able to detect l=1 modes that interpose
in the asymptotic period sequences and that provide a clear indication of mode
trapping in a stratified envelope, as predicted by theoretical models. A clear
signal is also present in the Kepler photometry at 3.387 d. Spectroscopic
observations reveal a radial-velocity amplitude of 64.8 km/s. We find that the
radial-velocity variations and the photometric signal have phase and amplitude
that are perfectly consistent with a Doppler-beaming effect and conclude that
the unseen companion, KIC 10553698B, must be a white dwarf most likely with a
mass close to 0.6 solar masses.
