  Drake and Szekeres have extended the Newman-Janis algorithm to produce
stationary axisymmetric spacetimes from general static spherically symmetric
solutions of the Einstein equations. The algorithm mathematically generates an
energy-momentum tensor for the rotating solution, but the rotating and
nonrotating system may or may not represent the same physical system, in the
sense of both being a perfect fluid, or an electromagnetic field, or a
$\Lambda$-term, and so on. In Part I (arxiv:2104.02255), we compared the
structure of the eigenvalues and eigenvectors of the rotating and nonrotating
energy-momentum tensors (their Segre types) and looked for the existence of
equations of state relating the rotating energy density and principal pressures
for Kerr-Schild systems. Here we extend our analysis to general static
spherically symmetric systems obtained according to the Drake-Szekeres
generalization of the Newman-Janis algorithm. We find that these rotating
systems can have almost all Segre types except [31] and [(31)]. Moreover, the
Segre type of the spacetime can change severely in passing from the nonrotating
to the rotating configurations, for example to $[11Z\bar{Z}]$ from seed systems
which were initially [(111,1)]. We also find conditions dictating how many
equations of state may exist in a Drake-Szekeres system.
