Half-vortices have been recently shown to be the elementary topological defects supported by a spinor cavity exciton-polariton condensates with spin-anisotropic interactions [Y. G. Rubo, Phys. Rev. Lett. 99, 106401 (2007)]. A half-vortex is composed by an integer vortex for one circular component of the condensate, whereas the other component remains static. We analyze theoretically the effect of the splitting between TE and TM polarized eigen modes on the structure of the vortices in this system. For TE and TM modes, the polarization states depend on the direction of propagation of particles and impose some well-defined phase relation between the two circular components. As a result elementary topogical defects in this system are no more half-vortices but integer vortices for both circular components of the condensate. The intrinsic lifetime of half-vortices is given and the texture of a few vortex states is analyzed.