  Anisotropic flow of kaons and antikaons is studied in heavy-ion collisions at
CERN SPS and BNL RHIC energies within the microscopic quark-gluon string model.
In the midrapidity range the directed flow of kaons v_1 differs considerably
from that of antikaons at SPS energy (E_{lab} = 160 AGeV), while at RHIC energy
(\sqrt{s} = 130 AGeV) the excitation functions of both, kaon and antikaon,
flows coincide within the statistical error bars. The change is attributed to
formation of dense meson-dominated matter at RHIC, where the differences in
interaction cross-sections of kaons and antikaons become unimportant. The time
evolution of the kaon anisotropic flow is also investigated. The elliptic flow
of these hadrons is found to develop at midrapidity at times 3 < t < 10 fm/c,
which is much larger than the nuclear passing time t^{pass} = 0.12 fm/c. As a
function of transverse momentum the elliptic flow increases almost linearly
with rising p_t. It stops to rise at p_t > 1.5 GeV/c reaching the saturation
value $v_2^K (p_t) \approx 10%$.
