Applications of Lie groups to the theory of equilibrium of cylindrically symmetric magnetic flux tubes

A group analysis of differential equations for the self-consistent equilibrium of cylindrically symmetric magnetic flux tubes is presented. The existence of non-Maxwellian distribution functions, which make it possible to find analytical expressions for magnetic field and plasma density, is proved. The distribution functions are assumed to depend only on constants of motion, which makes it possible to formulate and solve the group classification problem for nonlinear third-order differential equations. Analytical solutions to differential equations with a nontrivial symmetry group are calculated. The results can be used to analyze fractal properties of plasma configurations in solar and stellar atmospheres for different distribution functions.