This paper presents some analytical results related to the determination of the singular poses of the 3-RPS parallel manipulator at which it gains two degrees of freedom. The forward kinematic univariate (FKU) of the manipulator acquires a special structure at such a pose. All such poses have been identified in the closed-form, using a Stüdy-parameter representation of SE(3), for both the operation modes of the 3-RPS. These results are novel, to the best of the knowledge of the authors, and these have been verified using the traditional method, using the criterion of loss of rank of certain Jacobian matrices. The theoretical results have been illustrated with numerical examples.