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, $T_cpropto (B_c-B)^{2/3}$. 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.
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