  The Newman-Janis algorithm and its generalizations can be used mathematically
to generate rotating solutions from nonrotating spherically-symmetric solutions
within general relativity. The energy-momentum tensors of these solutions 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
a series of two papers, we compare the structure of the eigenvalues and
eigenvectors of the rotating and nonrotating energy-momentum tensors (their
Segre types) and look for the existence of equations of state relating the
energy density and the principal pressures. Part I covers Kerr-Schild systems,
Part II more general systems.
  We find that there is a unique family of stationary axisymmetric Kerr-Schild
systems that obey the same equation of state in both the rotating and
nonrotating configurations. This family includes the Kerr and Kerr-Newman black
holes, as well as rotating spacetimes whose mass function in the nonrotating
limit contains a constrained superposition of a cloud of strings term, a
Reissner-Nordstrom term, a cosmological constant term, and a Schwarzschild
term. We describe the common equation of state relating energy density and
pressure in this family of spacetimes and discuss some of its properties.
