Based on the Lie-group-algebraic properties of the displacement set, the 4DOF primitive generators of the Schoenflies motion termed -motion for brevity are briefly recalled. An -motion includes 3DOF spatial translation and any 1DOF rotation provided that the axes are parallel to a given direction. The serial concatenation of two generators of 4DOF -motion produces a 5DOF motion called double Schoenflies motion or -motion for brevity, which includes 3DOFs of translations and any 2DOFs of rotations if the axes are parallel to two independent vectors. This is established using the composition product of two Lie subgroups of -motion. All possible 5DOF serial chains with distinct general architectures for the generation of -motion are comprehensively introduced in the beginning. The parallel setting between a fixed base and a moving platform of two 5DOF limbs, under particular geometric conditions, makes up a 4DOF isoconstrained parallel generator (abbreviated as ) of a Schoenflies motion set. “Isoconstrained” is synonymous with “nonoverconstrianed,” and the corresponding chains are trivial chains of the 6D group of general 6DOF motions and can move in the presence of manufacturing errors. Moreover, related families of s are also deducted by using the reordering or the commutation of the factor method, which yields more 5D subsets of displacements containing also the -motion of the end effector. In that way, several novel general-type architectures of 4DOF parallel manipulators with potential applications are synthesized systematically in consideration of the actuated pairs near the fixed base.