For the energy transportation from a generator to a load, vacuum transmission lines with the magnetic insulation (MITL) are used, where the nonlinear electromagnetic waves can be formed because of both nonlinear effects and the electron leakage at the wave front. By the plasma switch experiments, the vacuum gap of the MITL is filled by a low density plasma, so the problem arises about the dynamics of electromagnetic fields in the plasma medium. There exists also some another reason for the plasma appearance in the MITL-The ion emission from the anode surface. In the following the plasma will be considered in the framework of the two-fluid magnetohydrodynamics (MHD) model by the account of the Hall effect, when electrons are magnetized and ions are not magnetized. In the absence of interparticle collisions, the electron vorticity equal to the curl of the generalized momentum of electron is conserved. It is easy to verify that the ratio of the electron vorticity to the electron density is conserved along electron trajectory, i.e.This value proves to be a Lagrange invariant. In addition, we consider the turbulent plasma medium, as it is usual for the plasma opening switch device. Then the Lagrange invariant is equalized in space because of the turbulent motion of electrons. The constant value of the Lagrange invariant allows to integrate the equations obtained. In the present paper, the propagation of the ion-diode-like equilibrium in the direction from a generator to a load is investigated.