Motivated by the problem of synthesizing a pattern of flexures that provide a desired constrained motion, this paper presents a new screw theory that deals with “line screws” and “line screw systems.” A line screw is a screw with a zero pitch. The set of all line screws within a screw system is called a line variety. A general screw system of rank is a line screw system if the rank of its line variety equals . This paper answers two questions: (1) how to calculate the rank of a line variety for a given screw system and (2) how to algorithmically find a set of linearly independent lines from a given screw system. It has been previously found that a wire or beam flexure is considered a line screw, or more specifically a pure force wrench. By following the reciprocity and definitions of line screws, we have derived the necessary and sufficient conditions of line screw systems. When applied to flexure synthesis, we show that not all motion patterns can be realized with wire flexures connected in parallel. A computational algorithm based on this line screw theory is developed to find a set of admissible line screws or force wrenches for a given motion space. Two flexure synthesis case studies are provided to demonstrate the theory and the algorithm.