  Cosmic Birefringence (CB) is a phenomenon in which the polarization of the
Cosmic Microwave Background (CMB) radiation is rotated as it travels through
space due to the coupling between photons and an axion-like field. We look for
a solution able to explain the result obtained from the \textit{Planck} Public
Release 4 (PR4), which has provided a hint of detection of the CB angle,
$\alpha=(0.30\pm0.11)^{\circ}$. In addition to the solutions, already present
in the literature, which need a non-negligible evolution in time of the
axion-like field during recombination, we find a new region of the parameter
space which allows for a nearly constant time evolution of such a field in the
same epoch. The latter reinforces the possibility to employ the commonly used
relations connecting the observed CMB spectra with the unrotated ones, through
trigonometric functions of the CB angle. However, if the homogeneous axion
field sourcing isotropic birefringence is almost constant in time during the
matter-dominated era, this does not automatically implies that the same holds
true also for the associated inhomogeneous perturbations. For this reason, in
this paper we present a full generalized Boltzmann treatment of this
phenomenon, that is able, for the first time to our knowledge to deal with the
time evolution of anisotropic cosmic birefringence (ACB). We employ this
approach to provide predictions of ACB, in particular for the set of best-fit
parameters found in the new solution of the isotropic case. If the latter is
the correct model, we expect an ACB spectrum of the order of
$(10^{-15}\div10^{-32})$ deg$^2$ for the auto-correlation, and
$(10^{-7}\div10^{-17})$ $\mu $K$\cdot\,$deg for the cross-correlations with the
CMB $T$ and $E$ fields, depending on the angular scale.
