A 5-SPU robot with collinear universal joints is well suited to handling an axisymmetric tool, since it has 5 controllable DoFs and the remaining one is a free rotation around the tool. The kinematics of such a robot having also coplanar spherical joints has previously been studied as a rigid subassembly of a StewartGough platform, it being 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 3D space of leg attachments, we prove that there are only three general topologies for the singularity locus corresponding to the families of quartically-, cubicallyand quadratically-solvable 5-SPU robots. The members of the last family have only 4 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.