Noether symmetries and stability of Kantowski–Sachs spacetimes via geodesic deviation

This paper aims to explore Noether symmetries associated with the Lagrangian of the Kantowski–Sachs spacetime metric via Rif tree technique. Rather than using the traditional method of directly integrating the symmetry equations, we have implemented a MAPLE algorithm that systematically simplifies these equations into a reduced involutive form (Rif). This approach produces a variety of metrics that possess different dimensional Noether algebras, organized as branches in a structure called a Rif tree. We have solved the symmetry equations for each metric to extract the vector fields that generate the desired Noether symmetries of dimensions 5, 6, 7, 8, 9, and 11. Our analysis reveals that the Rif tree method provides new insights that cannot be achieved through the conventional direct integration. Furthermore, we have assessed the stability of the resulting metrics by utilizing the geodesic deviation equations obtained from perturbations of the geodesic equations. We have also included a brief discussion of energy conditions satisfied by the metrics we derived.