  We use the Renormalization Group method to study the magnetic field influence
on the Bose-Einstein condensation of interacting dilute magnons in three
dimensional spin systems. We first considered a model with SU(2) symmetry
(universality class $z=1$) and we obtain for the critical magnetic field a
power law dependence on the critical temperature, $[H_c(T)-H_c(0)]\sim T^2$. In
the case of U(1) symmetry (universality class $z=2$) the dependence is
different, and the magnetic critical field depends linearly on the critical
temperature, $[H_c(T)-H_c(0)]\sim T$. By considering a more relevant model,
which includes also the system's anisotropy, we obtain for the same symmetry
class a $T^{3/2}$ dependence of the magnetic critical field on the critical
temperature. We discuss these theoretical predictions of the renormalization
group in connection with experimental results reported in the literature.
