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

In $3\times 3$ dimensions, entangled mixed states that are positive under partial transposition (PPT states) must have rank at least four. They are well understood. We say that they have rank $(4,4)$ since a state $\rho$ and its partial transpose $\rho^P$ both have rank four. The next problem is to understand the extremal PPT states of rank $(5,5)$. We call two states $\textrm{SL}\otimes\textrm{SL}$-equivalent if they are related by a product transformation. 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 product vectors in their kernels. The three numbers $\{6,6;0\}$ are $\textrm{SL}\otimes\textrm{SL}$-invariants that help us classify the state. We have studied numerically a few types of nongeneric rank five PPT states, in particular states with one or more product vectors in their kernels. We find an interesting new analytical construction of all rank four extremal PPT states, up to $\textrm{SL}\otimes\textrm{SL}$-equivalence, where they appear as boundary states on one single five dimensional face on the set of normalized PPT 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 a special form of $\textrm{SL}\otimes\textrm{SL}$-symmetry. We have produced numerically a random sample of rank $(5,5)$ $\textrm{SL}\otimes\textrm{SL}$-symmetric states. About fifty of these are of type $\{6,6;0\}$, among those all are extremal and about half are genuinely $\textrm{SL}\otimes\textrm{SL}$-symmetric.