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A 5-SPU robot with collinear universal joints is well suited to handle an axisymmetric tool, since it has five controllable degrees of freedom, and the remaining one is a free rotation around the tool. The kinematics of such a robot also having coplanar spherical joints has previously been studied as a rigid subassembly of a Stewart-Gough platform, which has been denoted a line-plane component. Here, we investigate how to move the leg attachments in the base and the platform without altering the robot's singularity locus. By introducing the so-called 3-D space of leg attachments, we prove that there are only three general topologies for the singularity locus corresponding to the families of quartically, cubically, and quadratically solvable 5-SPU robots. The members of the last family have only four assembly modes, which are obtained by solving two quadratic equations. Two practical features of these quadratically solvable robots are the large manipulability within each connected component and the fact that, for a fixed orientation of the tool, the singularity locus reduces to a plane.