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In a recent series of papers new exact analytical solutions of Einsteinequations representing interior spacetimes sourced by stationary rigidlyrotating cylinders of fluids have been displayed. We have first considered afluid 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 anisotropicpressure is radially oriented, J. Math. Phys. 64, 052502 (2023). This work isbeing currently extended to the cases of differentially rotating irrotationalfluids. The results are presented in a new series of papers considering, inturn, a perfect fluid source, arXiv:2305.11565 [gr-qc], and the same threeanisotropic pressure cases. Here, fluids with an axially directed pressure areconsidered. A general method for generating new mathematical solutions to thefield equations is displayed and three classes are presented so as to exemplifythis recipe. Their mathematical and physical properties are analyzed. The firstclass, named class A, whose other mathematical and physical propertiesdetermine a standard configuration, is shown to exhibit a singular axis ofsymmetry which can be considered as an awkward drawback. The second class,class B, is free from such a singularity but appears to exhibit a negativeenergy density which characterizes a rather exotic kind of matter. The thirdclass, class C, is the best behaved since it possesses the main propertiesexpected from spacetimes sourced by rather standard fluids. The three classesare matched to an exterior Lewis-Weyl vacuum and the conditions for avoiding anangular deficit are discussed. A comparison with the rigidly rotating fluidcase is provided.

Interior spacetimes sourced by stationary differentially rotating irrotational cylindrical fluids. II. Axial pressure