Form Invariance and Lie Symmetries of the Rotational Relativistic Birkhoff System

For a rotational relativistic Birkhoff system, the relation between the form invariance and the Lie symmetries are given under infinitesimal transformations of groups. If the infinitesimal transformation generators ξ₀ and ξ_μ satisfy the conditions of the form invariance, and the determining equation of Lie symmetries holds, the form invariance leads to a Lie symmetry of the system. Furthermore, if the infinitesimal transformations generators ξ₀ and ξ_μ satisfy the conditions of the form invariance and the determining equation of Lie symmetry holds, and if there exists a gauge function G satisfying the structure equation of Lie symmetry, then the form invariance will lead to the Lie symmetrical conserved quantity of the system. An example is given to illustrate the application of the results.