The analysis of possible current distributions when passing current through a periodically modulated long Josephson contact located in an external magnetic field is carried out. An approach based on the analysis of continuous configuration modification proceeding in the direction of Gibbs potential reduction is used for the calculation. The case when the pinning parameter is less than the critical value is considered. It is shown that at any value of the external magnetic field, there is a critical value of the transport current, when exceeded, the situation ceases to be stationary, as a result of which energy passes into radiation and heat, i.e. currents cease to be persistent. The value of the critical current is determined by the value of the magnetic field at which the vortices begin to fill the entire length of the contact. With an increase in the external magnetic field, the critical value of the current decreases.