  We consider low rank density operators $\varrho$ supported on a $M\times N$
Hilbert space for arbitrary $M$ and $N$ ($M\leq N$) and with a positive partial
transpose (PPT) $\varrho^{T_A}\ge 0$. For rank $r(\varrho) \leq N$ we prove
that having a PPT is necessary and sufficient for $\varrho$ to be separable; in
this case we also provide its minimal decomposition in terms of pure product
states. It follows from this result that there is no rank 3 bound entangled
states having a PPT. We also present a necessary and sufficient condition for
the separability of generic density matrices for which the sum of the ranks of
$\varrho$ and $\varrho^{T_A}$ satisfies $r(\varrho)+r(\varrho^{T_A}) \le
2MN-M-N+2$. This separability condition has the form of a constructive check,
providing thus also a pure product state decomposition for separable states,
and it works in those cases where a system of couple polynomial equations has a
finite number of solutions, as expected in most cases.
