To solve for the Lie symmetries of a holonomic systems with redundant coordinates, the GeM package of the Maple is used to analytically solve the determining equations for the Lie symmetries of the system. This process yields the maximally linearly independent set of Lie symmetry solutions for the system. The holonomic constraints are formulated as integrable velocity constraints. For the purpose of reduction, two different fiber structures are constructed. We reduce the system onto a space that simultaneously satisfies both the constraint distribution and the symmetry vector field (the tangent vectors to the orbits of the Lie symmetry group). A symmetry reduction for the holonomic system is achieved, and corresponding numerical algorithm is provided. An example is given to illustrate the application of the results.