This paper deals with the characterization of the operation modes of the 2-RUU parallel manipulator with an algebraic approach, namely the Study kinematic mapping of the Euclidean group SE(3). The manipulator is described by a set of eight constraint equations and the primary decomposition reveals that the mechanism has three operation modes. The singularity conditions are obtained by deriving the determinant of the Jacobian matrix of the constraint equations with respect to the Study parameters. It is shown that there exist singular configurations in which the 2-RUU manipulator may switch from one operation mode to another operation mode. All the singular configurations are mapped onto the joint space and are geometrically interpreted. Finally, the mechanism may switch from the 1st Schönflies mode to the 2nd Schönflies mode through the additional mode that contains self-motions.