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This paper discusses a generalized model for compact stars, assumed to be anisotropic in nature due to the presence of highly dense and ultra-relativistic matter distribution. After embedding the 4D Riemannian space locally and isometrically into a 5D pseudo-Euclidean space, we solve the Einstein equations by employing a class of physically acceptable metric functions. The physical properties determined include the anisotropic factor showing that the anisotropy is zero at the center and maximum at the surface. Other boundary conditions yield the values of various parameters needed for rendering the numerous plots and also led to the EOS parameters. It is further determined that the usual energy conditions are satisfied and that the compact structures are stable, based on several criteria, viz., the equilibrium of forces, Herrera cracking concept, adiabatic index, etc. We note that the proposed stellar model satisfies the Buchdahl condition. Finally, the values of the numerous constants and physical parameters are determined, specifically for the compact stellar object LMC X − 4 , which we choose as a representative of the compact stars to present the analysis of the obtained results. Finally, we show that the present generalized model can justify most of the compact stars including white dwarfs and ultra-dense compact stars for a suitable tuning of the parametric values of n. [ABSTRACT FROM AUTHOR]