  In recent years, several close post-common-envelope eclipsing binaries have
been found to show cyclic eclipse timing variations (ETVs). This effect is
usually interpreted either as the gravitational interaction among circumbinary
bodies and the host binary -- known as the light travel time (LTT) effect -- or
as the quadrupole moment variations in one magnetic active component -- known
as Applegate mechanism. In this study, we present an analysis of the ETV and
the magnetic cycle of the close binary KIC 10544976. This system is composed of
a white dwarf and a red dwarf in a short orbital period (0.35 days) and was
monitored by ground-based telescopes between 2005 and 2017 and by the Kepler
satellite between 2009 and 2013. Using the Kepler data, we derived the magnetic
cycle of the red dwarf by two ways: the rate and energy of flares and the
variability due to spots. Both methods resulted in a cycle of ~600 days, which
is in agreement with magnetic cycles measured for single low-mass stars. The
orbital period of KIC 10544976 shows only one long-term variation which can be
fitted by an LTT effect with period of ~16.8 yr. Hence, one possible
explanation for the ETVs is the presence of a circumbinary body with minimal
mass of ~13.4 M_Jup. In the particular scenario of coplanarity between the
external body and the inner binary, the third body mass is also ~13.4 M_Jup. In
this case, the circumbinary planet must either have survived the evolution of
the host binary or have been formed as a consequence of its evolution.
