  In this paper, we address the problem of weak solutions of Yudovich type for
the inviscid MHD equations in two dimensions. The local-in-time existence and
uniqueness of these solutions sound to be hard to achieve due to some terms
involving Riesz transforms in the vorticity-current formulation. We shall prove
that the vortex patches with smooth boundary offer a suitable class of initial
data for which the problem can be solved. However this is only done under a
geometric constraint by assuming the boundary of the initial vorticity to be
frozen in a magnetic field line.
  We shall also discuss the stationary patches for the incompressible Euler
system $(E)$ and the MHD system. For example, we prove that a stationary simply
connected patch with rectifiable boundary for the system $(E)$ is necessarily
the characteristic function of a disc.
