  Based on the so-called re-scaling method, we will give a detailed description
of the solutions to the Hamiltonian system (\ref{Hsystem}) below, which was
discovered only recently by Kecker, and is strongly related to Painleve's
fourth differential equation. In particular, the problem to determine those
fourth Painleve transcendents with positive Nevanlinna deficiency
$\delta(0,w)$, is completely resolved.
