  The existence of global-in-time classical solutions to the Cauchy problem of
incompressible Magnetohydrodynamic flows with zero magnetic diffusivity is
considered in two dimensions. The linearization of equations is a degenerated
parabolic-hyperbolic system. The solution is constructed as a small
perturbation of a constant background in critical spaces. The deformation
gradient has been introduced to decouple the subtle coupling between the flow
and the magnetic field. The $L^1$ dissipation of the velocity is obtained.
