  This article is the second in a series devoted to the study of spacetimes
sourced by a stationary cylinder of fluid rigidly rotating around its symmetry
axis and exhibiting an anisotropic pressure by using new exact interior
solutions of General Relativity. The configurations have been specialized to
three different cases where the pressure is on turn directed alongside each
principal stress. The two first articles in the series display the analysis of
the axial pressure case. Indeed, the first axial class published in Paper 1 is
merely a special case. It is recalled here and its properties are revised and
supplemented. Moreover, a fully general method aiming at constructing different
classes of such solutions is displayed. This method described in the present
paper, Paper 2, represents a key result of this work. It is exemplified and
applied to two new classes of solutions depending on a single constant
parameter. One of them, denoted Class A, is shown to verify every conditions
needing to be satisfied by a fully achieved set of exact solutions: axisymmetry
and regularity conditions, matching to an exterior vacuum, proper metric
signature, weak and strong energy conditions. Other properties and general
rules are exhibited, some shedding light on rather longstanding issues.
Astrophysical and physical applications are suggested.
