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The present work deals with scalar field cosmology in the framework of aquantum gravity modified Einstein-Hilbert Lagrangian with variable $G$ and$\Lambda$. Using Renormalization group, variable $G$ behaves as a minimallycoupled filed (not the scalar-tensor theory) and variable $\Lambda$ can beinterpreted as a potential function. The point Lagrangian for this model in thebackground of homogeneous and isotropic flat FLRW space-time model experiencespoint-like Noether symmetry and equivalent potential function $\Lambda(G)$ isdetermined. Using a point transformation in the $3D$ augmented space is foundthat one of the variable become cyclic and as a consequence there isconsiderable simplification to the physical system. Lastly, the constants ofmotion can be written in compact form and it is possible to have analyticcosmological solutions in the present context.

Variable $G$ and $\Lambda$ gravity theory and analytical Cosmological Solutions using Noether symmetry approach