Conformal symmetries of Strumia-Tetradis' metric

In a recent paper (A. Strumia and N. Tetradis, JHEP 09 (2022) 203) a new conformally flat metric was introduced, describing an expanding scalar field in a spherically symmetric geometry. The spacetime can be interpreted as a Schwarzschild-like model with an apparent horizon surrounding the curvature singularity. For the above metric, we present the complete conformal Lie algebra consists of a six-dimensional subalgebra of isometries (Killing Vector Fields or KVFs) and nine proper conformal vector fields (CVFs). An interesting aspect of our findings is that there exist a gradient (proper) conformal symmetry (i.e. its bivector F_{ab} vanishes) which verifies the importance of gradient symmetries in constructing viable cocmological models with sound physical interest. In addition, the 9-dimensional conformal algebra imply the existence of constants of motion along null geodesics that allow us to determine the complete solution of null geodesic equation.