Parallel manipulators which are singular with respect to the Schönflies motion group X(a) are called Schönflies-singular, or more precisely X(a)-singular, where a denotes the rotary axis. A special class of such manipulators are architecturally singular ones because they are singular with respect to any Schönflies group. Another remarkable set of Schönflies-singular planar parallel manipulators of Stewart Gough type was already presented by the author in . Moreover the main theorem on these manipulators was given in . In this paper we give a complete discussion of the remaining special cases which also include so-called Cartesian-singular planar manipulators as side-product.