comparison with the asymptotically flat Kerr spacetimes, where the effect of spacetime rotation vanishes for asymptotically large values of the radius, in the KdS spacetimes the properties of the circular orbits must be treated more carefully, because the rotational effect is relevant in the whole region where the circular orbits are allowed and it survives even at the cosmological horizon.

The minus-family orbits have $L_-<0$ in each KdS spacetime and such orbits are counter-rotating relative to the LNRF.

In the black-hole spacetimes, the plus-family orbits are co-rotating in almost all radii where the circular orbits are allowed except some region in vicinity of the static radius, where they become to be counter-rotating. However, these orbits are unstable. The specific angular momentum and energy of particles located at the static radius, where the plus-family and minus-family orbits coalesce, are given by $$L_{\mathrm{s}} = -a\frac{3y^{1/3}+a^{2}y}{\left (1-3y^{1/3}-a^{2}y
  \right )^{1/2}}, \quad
E_{\mathrm{s}} = (1-3y^{1/3}-a^2 y)^{1/2}.$$

In the naked-singularity spacetimes, the plus-family orbits behave in a more complex way. They are always counter-rotating in vicinity of the static radius. Moreover, in the naked singularity spacetimes with the rotational parameter low enough ($a<3\sqrt{3}/4,\ y=0$), stable counter-rotating plus-family circular orbits exist. When $a$ is very close to 