  In an important series of articles published during the 70's, Krasi\'nski
displayed a class of interior solutions of the Einstein field equations sourced
by a stationary isentropic rotating cylinder of perfect fluid. However, these
solutions depend on an unspecified arbitrary function, which lead the author to
claim that the equation of state of the fluid could not be obtained directly
from the field equations but had to be added by hand. In the present article,
we use a double ansatz which we have developed in 2021 and implemented at
length into a series of recent papers displaying exact interior solutions for a
stationary rotating cylindrically symmetric fluid with anisotropic pressure.
This ansatz allows us to obtain here a fully integrated class of solutions to
the Einstein equations, written with the use of very simple analytical
functions, and to show that the equation of state of the fluid follows
naturally from these field equations.
