Group analysis of structure equations for stars in radiative and convective equilibrium

It is proposed to use the Lie group theory of symmetries of differential equations to investigate the system of equations describing a static star in a radiative and convective equilibrium. It is shown that the action of an admissible group induces a certain algebraic structure in the set of all solutions, which can be used to find a family of new solutions. We have demonstrated that, in the most general case, the equations admit an infinite parameter group of quasi-homologous transformations. We have found invariants of the symmetries group which correspond to the fundamental relations describing a physical characteristic of the stars such as the Hertzsprung-Russell diagram or the mass-luminosity relation. In this way we can suggest that group invariants have not only purely mathematical sense, but their forms are closely associated with the basic empirical relations.