Lie symmetries and conserved quantities of relativistic Birkhoff systems

The Pfaff-Birkhoff principle and the Birkhoff equations for relativistic Bi rkhoff systems are given. The definition of an infinitesimal generator for the relativistic Birkhoff systems is given; by using the invariance of the ordinary differential equations under the infinitesimal transformations, th e determining equations of Lie symmetries for the systems are established; the structure equation and the conserved quantities are obtained. And the i nverse problem of Lie symmetries of the systems is also studied. Two exampl es are given to illustrate the application of the result.