  In this paper, we investigate the Cauchy problem of the nonhomogeneous
incompressible non-resistive MHD on $\mathbb{R}^2$ with vacuum as far field
density and prove that the 2D Cauchy problem has a unique local strong solution
provided that the initial density and magnetic field decay not too slow at
infinity. Furthermore, if the initial data satisfy some additional regularity
and compatibility conditions, the strong solution becomes a classical one.
