A group analysis of a system describing an ideal plastic flow is made in order to obtain analytical solutions. The complete Lie algebra of point symmetries of this system are given. Two of the infinitesimal generators that span the Lie algebra are original to this paper. A classification into conjugacy classes of all one- and two-dimensional subalgebras is performed. Invariant and partially invariant solutions corresponding to certain conjugacy classes are obtained using the symmetry reduction method. Solutions of algebraic, trigonometric, inverse trigonometric and elliptic type are provided as illustrations and other solutions expressed in terms of one or two arbitrary functions have also been found. For some of these solutions, a physical interpretation allows one to determine the shape of feasible extrusion dies corresponding to these solutions. The corresponding tools could be used to curve rods or slabs, or to shape a ring in an ideal plastic material by an extrusion process.