  We address the group classification problem for gravitational field equations
within the context of brane-world cosmology, considering the presence of a bulk
scalar field. Our investigation revolves around a five-dimensional spacetime,
with the four-dimensional Friedmann--Lema\^{\i}tre--Robertson--Walker geometry
embedded within it. Additionally, we assume that the scalar field exists in
this five-dimensional geometry (bulk) and possesses a nonzero mass. The
resulting field equations constitute a system of nonlinear partial differential
equations. We apply the Lie symmetry condition to identify all functional forms
of the scalar field potential, ensuring that the field equations remain
invariant under one-parameter point transformations. Consequently, we find that
only the exponential potential exhibits Lie symmetries. Finally, the Lie
invariants are used to construct similarity transformations, which enable us to
derive exact solutions for the system.
