  Phase covariant qubit dynamics describes an evolution of a two-level system
under simultaneous action of pure dephasing, energy dissipation, and energy
gain with time-dependent rates $\gamma_z(t)$, $\gamma_-(t)$, and $\gamma_+(t)$,
respectively. Non-negative rates correspond to completely positive divisible
dynamics, which can still exhibit such peculiarities as non-monotonicity of
populations for any initial state. We find a set of quantum channels attainable
in the completely positive divisible phase covariant dynamics and show that
this set coincides with the set of channels attainable in semigroup phase
covariant dynamics. We also construct new examples of eternally indivisible
dynamics with $\gamma_z(t) < 0$ for all $t > 0$ that is neither unital nor
commutative. Using the quantum Sinkhorn theorem, we for the first time derive a
restriction on the decoherence rates under which the dynamics is positive
divisible, namely, $\gamma_{\pm}(t) \geq 0$, $\sqrt{\gamma_+(t) \gamma_-(t)} +
2 \gamma_z(t) > 0$. Finally, we consider phase covariant convolution master
equations and find a class of admissible memory kernels that guarantee complete
positivity of the dynamical map.
