Current geometric path planners for nonholonomic parallel orienting robots generate maneuvers consisting of a sequence of moves connected by zero-velocity points. The need for these maneuvers restrains the use of this kind of parallel robots to few applications. Based on a rather old result on linear time-varying systems, this letter shows that there are infinitely differentiable paths connecting two arbitrary points in SO(3) such that the instantaneous axis of rotation along the path rest on a fixed plane. This theoretical result leads to a practical path planner for nonholonomic parallel orienting robots that generates single-move maneuvers. To present this result, we start with a path planner based on three-move maneuvers, and then we proceed by progressively reducing the number of moves to one, thus providing a unified treatment with respect to previous geometric path planners. Peer Reviewed