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Three-dimensional anisotropic magneto-hydrodynamical system is investigated in the whole space ${ℝ}^3$. Existence and uniqueness results are proved in the anisotropic Sobolev space $H^{0,s}$ for $s\>1/2$. Asymptotic behavior of the solution when the Rossby number goes to zero is studied. The proofs, where the incompressibility condition is crucial, use the energy method, an appropriate dyadic decomposition of the frequency space, product laws in anisotropic Sobolev spaces and Strichartz-type estimates.