The model of rain-wind-induced vibration of continuous stay cable is built on the basis of the dAlembert principle. This model can describe the dynamic characteristics of continuous stay cable better than the truncated model. By analyzing the static bifurcation behaviors according to the singularity theory，we describe the bifurcation problem as a one-codimensional unfolding with Z2 symmetry. The transition set and bifurcation diagrams for the singularity are presented，while the stability of the zero solution is studied using the eigenvalues in various parameter regions. The corresponding relationship between the unfolding parameters and the physical parameters will facilitates further analysis of the bifurcation behaviors of the stay cable，and it also provides theoretical basis for the design of cable stayed bridge. Meanwhile，it is helpful to the development and application of the engineering projects.