The propagation of solitons in low-birefringent optical fiber is considered when four-wave-mixing leads to energy exchange between polarization components. The stability of stationary solutions is addressed with a linear stability analysis expressed as an eigenvalue equation. It is shown that in the vicinity of the bifurcation point the eigenvalues must be either pure real (stable) or pure imaginary (unstable). The transition between these (zero eigenvalue) lends itself to analytical solutions, in spite of the nonintegrability of the original system of partial differential equations. It is demonstrated that the marginally stable perturbations of phase shift and temporal shift also apply to mixed-mode stationary solutions. The bifurcation point is found exactly along with the corresponding eigenfunction. This analysis provides the elliptically polarized stationary solution just above the bifurcation point. It is also shown that the fast-mode soliton just below the bifurcation point and the elliptically polarized soliton just above the bifurcation point are stable.
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