  The lowest quadrupole $\gamma$-vibrational $K^{\pi}=2^+$ states in axially
deformed rare-earth (Nd, Sm, Gd, Dy, Er, Yb, Hf, W) and actinide (U) nuclei are
systematically investigated within the separable random-phase-approximation
(SRPA) based on the Skyrme functional. The energies $E_{\gamma}$ and reduced
transition probabilities $B(E2)$ of $2^+_{\gamma}$-states are calculated with
the Skyrme forces SV-bas and SkM$^*$. The energies of two-quasiparticle
configurations forming the SRPA basis are corrected by using the pairing
blocking effect. This results in a systematic downshift of $E_{\gamma}$ by
0.3-0.5 MeV and thus in a better agreement with the experiment, especially in
Sm, Gd, Dy, Hf, and W regions. For other isotopic chains, a noticeable
overestimation of $E_{\gamma}$ and too weak collectivity of
$2^+_{\gamma}$-states still persist. It is shown that domains of nuclei with a
low and high $2^+_{\gamma}$ -collectivity are related with the structure of the
lowest 2-quasiparticle states and conservation of the Nilsson selection rules.
The description of $2^+_{\gamma}$ states with SV-bas and SkM$^*$ is similar in
light rare-earth nuclei but deviates in heavier nuclei. However SV-bas much
better reproduces the quadrupole deformation and energy of the isoscalar giant
quadrupole resonance. The accuracy of SRPA is justified by comparison with
exact RPA. The calculations suggest that a further development of the
self-consistent calculation schemes is needed for a systematic satisfactory
description of the $2^+_{\gamma}$ states.
