  We analyze photometric data of the active RS CVn--type star HR 1099 for the
years 1975--2006 with an inversion technique and reveal the nature of two
activity cycles of 15--16 yr and 5.3$\pm$0.1 yr duration. The 16 yr cycle is
related to variations of the total spot area and is coupled with the
differential rotation, while the 5.3 yr cycle is caused by the symmetric
redistribution of the spotted area between the opposite stellar hemispheres
(flip-flop cycle). We recover long-lived active regions comprising two active
longitudes that migrate in the orbital reference frame with a variable rate
because of the differential rotation along with changes in the mean spot
latitudes. The migration pattern is periodic with the 16 yr cycle. Combining
the longitudinal migration of the active regions with a previously measured
differential rotation law, we recover the first stellar butterfly diagram
without an assumption about spot shapes. We find that mean latitudes of active
regions at opposite longitudes change antisymmetrically in the course of the 16
yr cycle: while one active region migrates to the pole, the other approaches
the equator. This suggests a precession of the global magnetic field with
respect to the stellar rotational axis.
