  In a recent series of papers new exact analytical solutions of Einstein
equations representing interior spacetimes sourced by stationary rigidly
rotating cylinders of fluids have been displayed. We have first considered a
fluid with an axially directed pressure C\'el\'erier, Phys. Rev. D 104, 064040
(2021), J. Math. Phys. 64, 032501 (2023), then a perfect fluid, J. Math. Phys.
64, 022501 (2023), followed by a fluid with an azimuthally directed pressure,
J. Math. Phys. 64, 042501 (2023), and finally a fluid where the anisotropic
pressure is radially oriented, J. Math. Phys. 64, 052502 (2023). This work is
being currently extended to the cases of differentially rotating irrotational
fluids. The results are presented in a new series of papers considering, in
turn, a perfect fluid source, arXiv:2305.11565 [gr-qc], and the same three
anisotropic pressure cases. Here, fluids with an axially directed pressure are
considered. A general method for generating new mathematical solutions to the
field equations is displayed and three classes are presented so as to exemplify
this recipe. Their mathematical and physical properties are analyzed. The first
class, named class A, whose other mathematical and physical properties
determine a standard configuration, is shown to exhibit a singular axis of
symmetry which can be considered as an awkward drawback. The second class,
class B, is free from such a singularity but appears to exhibit a negative
energy density which characterizes a rather exotic kind of matter. The third
class, class C, is the best behaved since it possesses the main properties
expected from spacetimes sourced by rather standard fluids. The three classes
are matched to an exterior Lewis-Weyl vacuum and the conditions for avoiding an
angular deficit are discussed. A comparison with the rigidly rotating fluid
case is provided.
