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Let E' denote the set of non-normalized two-qutrit entangled states of rankfour having positive partial transpose (PPT). We show that the set of SLOCCequivalence classes of states in E', equipped with the quotient topology, ishomeomorphic to the quotient R/A_5 of the open rectangular box R in theEuclidean space R^4 by an action of the alternating group A_5. We construct anexplicit map omega: Omega -> E', where Omega is the open positive orthant inR^4, whose image meets every SLOCC equivalence class E containeed in E'.Although the intersection of the image of omega and E is not necessarily asingleton set, it is always a finite set of cardinality at most 60. By abuse oflanguage, we say that any state in this intersection is a canonical form ofstates rho in E. In particular, we show that all checkerboard PPT entangledstates can be parametrized up to SLOCC equivalence by only two real parameters.We also summarize the known results on two-qutrit extreme PPT states and edgestates, and examine which other interesting properties they may have. Thus wefind the first examples of extreme PPT states whose rank is different from therank of its partial transpose.

Equivalence classes and canonical forms for two-qutrit entangled states of rank four having positive partial transpose