This paper deals with the constraint and the singularity analysis of the Exechon. Using the screw theory, the constraint and actuation wrenches acting on the moving platform are analyzed. The motion pattern of the Exechon is characterized based on a representation of the constraint wrenches in the projective space. A wrench graph representing the constraint and actuation wrench systems in the projective space is obtained. Based on this wrench graph, a superbracket of the Exechon is formulated. Finally, this superbracket is explored to provide the geometric conditions for the parallel singularities of the Exechon.