The authors consider the structure of the magnetic surfaces of a rectilinear stellarator containing a plasma through which a straight longitudinal current flows. The three cases of uniform, bell-shaped and parabolic current density distribution of a longitudinal current (cases of practical importance) are investigated.In the case of a uniform current distribution it is shown that if the rotational transform due to the current adds to that due to the vacuum field, the size of the separatrix decreases and at some critical value of the current, the separatrix shrinks to a point (resonance case).In the case of bell-shaped (or parabolic) radial current-density distribution, the magnetic surfaces of the stellarator can acquire a rosette structure. When the longitudinal current attains certain values, the separatrix is uncovered, i. e. the magnetic configuration is broken up. The conditions for the appearance of rosettes of a definite type and those for the break-up of magnetic surfaces are derived analytically.