Propagation of solitons in birefringent single-mode fibers near the zero-dispersion wavelength is considered. Initial pulses are assumed to be linearly polarized at an arbitrary angle with respect to the polarization axes. When the amplitudes of the partial pulses are equal, it is found that the third-order dispersion, as a small perturbation, does not affect the threshold of the soliton trapping. The evolution of two partial pulses is in a markedly different manner. The asymmetric oscillation structure is dominantly in one polarization. When the initial amplitudes are unequal, it is found that one or more soliton emerge from the initial pulse above a certain threshold. The two-captured pulses are very sharp and the continuum ends up in one polarization.