The differential equations of the second order autonomous generalized Birkhoff systems were firstly established. The linearized equations of this system were also given. The conditions for` the translation of linearized system into a gradient system were put forward. The singular points for linearized system were analyzed by the characteristic of the gradient system. Moreover, the types of singular points for the corresponding nonlinear system were studied by Perron theorem. The results show that if the linearized system can be translated into a gradient system, the singular point for the corresponding nonlinear system is probably a node or a saddle point.