Algebraic Solutions of Modified Field Equations in Anisotropic $$\boldsymbol{f(R,L_{m})}$$ Cosmology

We explore anisotropic cosmological models within the framework of modified gravity theory, specifically $$f(R,L_{m})$$ gravity, where the gravitational Lagrangian depends on both the Ricci scalar $$R$$ and the matter Lagrangian $$L_{m}$$ . We adopt the locally rotationally symmetric (LRS) Bianchi type I metric to investigate anisotropic cosmic evolution, which allows for directional dependence in the expansion rates while retaining analytical tractability. The modified field equations are derived using a particular form of the function $$f(R,L_{m})=R/2+L_{m}^{\alpha}$$ , where $$\alpha$$ quantifies the strength of the curvature-matter coupling. To obtain exact solutions, we consider power-law forms for the directional scale factors and study the cosmological dynamics under three distinct physically motivated assumptions on the equation of state: $$p+\rho=0$$ , $$\rho-p=0$$ , and $$\rho+3p=0$$ . In each case, the field equations are systematically reduced to algebraic forms and solved analytically. The resulting solutions exhibit both isotropic and anisotropic behavior, depending on the specific relationship between the expansion exponents and the coupling parameter $$\alpha$$ . For $$p+\rho=0$$ , the solutions suggest an effective cosmological constant-like behavior, with possible higher-dimensional interpretations where some dimensions remain static or compactified. Under the $$\rho-p=0$$ condition, a richer structure of solutions emerges, admitting both static, partially expanding, and fully anisotropic power-law evolutions, where anisotropy is directly controlled by the coupling parameter $$\alpha$$ . The analysis under $$\rho+3p=0$$ also yields anisotropic power-law solutions, where expansion and contraction coexist in different spatial directions. Overall, this work demonstrates the richness of cosmological behavior in $$f(R,L_{m})$$ gravity and highlights its potential to address key open problems in modern cosmology.