Classification of multipartite systems featuring only vertical bar W > and vertical bar GHZ > genuine entangled states

We present several multipartite quantum systems featuring the same type of genuine(tripartite) entanglement. Based on a geometric interpretation of the so-called vertical bar W > and vertical bar GHZ > states we show that the classification of all multipartite systems featuring those and only those two classes of genuine entanglement can be deduced from earlier work of algebraic geometers. In this paper, non-entangled pure states are identified with the highest weight orbit of the SLOCC group acting on the corresponding Hilbert space.