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In $3\times 3$ dimensions, entangled mixed states that are positive underpartial transposition (PPT states) must have rank at least four. They are wellunderstood. We say that they have rank $(4,4)$ since a state $\rho$ and itspartial transpose $\rho^P$ both have rank four. The next problem is tounderstand the extremal PPT states of rank $(5,5)$. We call two states$\textrm{SL}\otimes\textrm{SL}$-equivalent if they are related by a producttransformation. A generic rank $(5,5)$ PPT state $\rho$ is extremal, and $\rho$and $\rho^P$ both have six product vectors in their ranges, and no productvectors in their kernels. The three numbers $\{6,6;0\}$ are$\textrm{SL}\otimes\textrm{SL}$-invariants that help us classify the state. Wehave studied numerically a few types of nongeneric rank five PPT states, inparticular states with one or more product vectors in their kernels. We find aninteresting new analytical construction of all rank four extremal PPT states,up to $\textrm{SL}\otimes\textrm{SL}$-equivalence, where they appear asboundary states on one single five dimensional face on the set of normalizedPPT states. We say that a state $\rho$ is$\textrm{SL}\otimes\textrm{SL}$-symmetric if $\rho$ and $\rho^P$ are$\textrm{SL}\otimes\textrm{SL}$-equivalent, and is genuinely$\textrm{SL}\otimes\textrm{SL}$-symmetric if it is$\textrm{SL}\otimes\textrm{SL}$-equivalent to a state $\tau$ with$\tau=\tau^P$. Genuine $\textrm{SL}\otimes\textrm{SL}$-symmetry implies aspecial form of $\textrm{SL}\otimes\textrm{SL}$-symmetry. We have producednumerically a random sample of rank $(5,5)$$\textrm{SL}\otimes\textrm{SL}$-symmetric states. About fifty of these are oftype $\{6,6;0\}$, among those all are extremal and about half are genuinely$\textrm{SL}\otimes\textrm{SL}$-symmetric.

Nongeneric positive partial transpose states of rank five in $3\times 3$ dimensions