  In a recent series of papers new exact analytical solutions of the Einstein
equations representing interior spacetimes sourced by stationary rigidly
rotating cylinders of different kinds of fluids have been displayed, [Phys.
Rev. D {\bf 104}, 064040 (2021); J. Math. Phys. {\bf 64}, 022501 (2023); J.
Math. Phys. {\bf 64}, 032501 (2023); J. Math. Phys. {\bf 64}, 042501 (2023);
and J. Math. Phys. {\bf 64}, 052502 (2023)]. This work is currently being
extended to the cases of differentially rotating irrotational fluids. The
results are presented in a new series of papers considering in turn the same
three anisotropic pressure cases, as well as a perfect fluid source. Here, the
perfect fluid case is considered, and different classes are identified as
directly issuing from the field equations. Among them, an explicit analytical
set of solutions is selected as displaying perfect fluid spacetimes. Its
mathematical and physical properties are analyzed. Its matching to an exterior
Lewis-Weyl vacuum and the conditions for avoiding an angular deficit are
discussed.
