  We performed a comprehensive electron spin resonance, magnetization and heat
capacity study on the field-induced magnetic phase transitions in the kagome
antiferromagnet $\alpha-Cu_3Mg(OH)_6Br_2$. With the successful preparation of
single crystals, we mapped out the magnetic phase diagrams under the $c$-axis
and $ab$-plane directional magnetic fields $B$. For $B\|c$, the
three-dimensional (3D) magnon Bose-Einstein condensation (BEC) is evidenced by
the power law scaling of the transition temperature, $T_c\propto
(B_c-B)^{2/3}$. For $B\|ab$, the transition from the canted antiferromagetic
(CAFM) state to the fully polarized (FP) state is a crossover rather than phase
transition, and the characteristic temperature has a significant deviation from
the 3D BEC scaling. The different behaviors of the field-induced magnetic
transitions for $B\|c$ and $B\|ab$ could result from the Dzyaloshinkii-Moriya
(DM) interaction with the DM vector along the $c$-axis, which preserves the
$c$-axis directional spin rotation symmetry and breaks the spin rotation
symmetry when $B\|ab$. The 3D magnon BEC scaling for $B\|c$ is immune to the
off-stoichiometric disorder in our sample
$\alpha-Cu_{3.26}Mg_{0.74}(OH)_6Br_2$. Our findings have the potential to shed
light on the investigations of the magnetic anisotropy and disorder effects on
the field-induced magnon BEC in the quantum antiferromagnet.
