  This paper concerns the Cauchy problem of the nonhomogeneous incompressible
magnetohydrodynamic (MHD) equations on the whole two-dimensional (2D) space
with vacuum as far field density. In particular, the initial density can have
compact support. We prove that the 2D Cauchy problem of the nonhomogeneous
incompressible MHD equations admits a unique local strong solution provided the
initial density and the initial magnetic decay not too slow at infinity.
