The majority of the manipulation tasks require the motion of the end-effector from an initial pose to a final one without any particular condition on the path. Thus, the reduction of the practicable paths between any two poses is a possible choice, exploitable at the design stage, for simplifying the manipulator hardware. This choice is adopted in underactuated manipulators. The (nS)-2SPU wrist is one out of the underactuated parallel wrists this author proposed in a previous paper. Here, the kinematic analysis of this wrist is studied: both its finite and its elementary kinematics are considered. It is shown that its control algorithms can be written by using simple closed-form formulas, which can take advantage from the wide literature on the spherical four-bar linkages. Moreover, the demonstration that its singular configurations can be avoided more easily than the ones of the fully parallel wrist is provided.