  We study the modification of low temperature properties of quantum magnets
such as magnetization, heat capacity, energy spectrum, and densities of
condensed and noncondensed quasiparticles (triplons) due to anisotropies in the
framework of mean-field based approach. We show that in contrast to exchange
anisotropy (EA) interaction, Dzyaloshinsky-Moriya (DM) interaction modifies the
physics dramatically. Particularly, it changes the sign of the anomalous
density in the whole range of temperatures. Its critical behavior is slightly
modified also by the EA. We have found that the shift of the critical
temperature of phase transition (or crossover caused by DM interaction) is
positive and significant. Using the experimental data on the magnetization of
the compound TlCuCl$_3$, we have found optimal values for the strengths of EA
and DM interactions. The spectrum of the energy of low lying excitations has
also been investigated and found to develop a linear dispersion similar to
Goldstone mode with a negligibly small anisotropy gap.
