In this work, we have found a solution to the Einstein equations for a closed null string with axial symmetry, which radially increases its size. The possibility of choosing the ideal gas of null-strings as a dominant source of gravity is confirmed in the study of the null-string inflation mechanism in Friedman-Robertson-Walker spaces. The absence of a limit transition between the solution found and the solution for a closed null string in shape of a circle that changes its size radially is shown. This indicate a stability of configuration of a closed null string in shape of a circle during motion in an external gravitational field. It has been noted that a part of characteristics of null string gas, such as an ability to form a domain structure and an existence of stable polarized states (multi-string systems), does not depend on a shape of a null-string, but the dynamics of a test null string in a field of such a multi-string system will have peculiarities.
11.27.+d Extended classical solutions; cosmic strings, domain walls, texture
$^1$Crimean Federal University