  We present an analysis of the binary and physical parameters of a unique
pulsating white dwarf with a main-sequence companion, SDSS J1136+0409, observed
for more than 77 d during the first pointing of the extended Kepler mission: K2
Campaign 1. Using new ground-based spectroscopy, we show that this
post-common-envelope binary has an orbital period of 6.89760103(60) hr, which
is also seen in the photometry as a result of Doppler beaming and ellipsoidal
variations of the secondary. We spectroscopically refine the temperature of the
white dwarf to 12330(260) K and its mass to 0.601(36) Msun. We detect seven
independent pulsation modes in the K2 light curve. A preliminary asteroseismic
solution is in reasonable agreement with the spectroscopic atmospheric
parameters. Three of the pulsation modes are clearly rotationally split
multiplets, which we use to demonstrate that the white dwarf is not
synchronously rotating with the orbital period but has a rotation period of
2.49(53) hr. This is faster than any known isolated white dwarf, but slower
than almost all white dwarfs measured in non-magnetic cataclysmic variables,
the likely future state of this binary.
