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The present work shows the role of nonlinear saturation effects on modulational instability (MI) in a birefringent media with the help of nonlinear wave propagation technique. Using the Kundu–Eckhaus (KE) equation, we investigate both self phase and cross phase modulation effect on MI in a birefringent media without four wave mixing term. We consider only KE equation model without the existence of four wave mixing expression and the standard linear stability theory is presented to the appropriate model. In the case of modified KE equation, we discuss the role of MI in a birefringent fiber system with the help of conventional type of saturation effect. In the case of KE without four wave mixing, we observed that the SPM plays a vital role on MI in coupled birefringent fiber compared to XPM as shown in Ref. Parasuraman (2021). But, In saturable KE has different MI characteristic as well as growth rate of profile in birefringent model with the help of particular emphasis values of the nonlinear parameters. However, the nonlinear saturation KE model more helps to generate soliton or solitary wave in coupled nonlinear birefringent fiber in the presence of both SPM and XPM.