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We address the group classification problem for gravitational field equationswithin the context of brane-world cosmology, considering the presence of a bulkscalar field. Our investigation revolves around a five-dimensional spacetime,with the four-dimensional Friedmann--Lema\^{\i}tre--Robertson--Walker geometryembedded within it. Additionally, we assume that the scalar field exists inthis five-dimensional geometry (bulk) and possesses a nonzero mass. Theresulting field equations constitute a system of nonlinear partial differentialequations. We apply the Lie symmetry condition to identify all functional formsof the scalar field potential, ensuring that the field equations remaininvariant under one-parameter point transformations. Consequently, we find thatonly the exponential potential exhibits Lie symmetries. Finally, the Lieinvariants are used to construct similarity transformations, which enable us toderive exact solutions for the system.

Lie symmetries for the cosmological field equations in brane-world gravity with bulk scalar field