Schwarzschild–de Sitter spacetimes.[@Stu-Hle:1999:PHYSR4:] A “free” or “geodetical” observer on the static radius has only $U^t$ component of 4-velocity being non-zero. The position on the static radius is unstable relative to radial perturbations. The second restriction is given by the condition $$\label{e50}
        1 -\frac{3}{r} -a^2y \pm 2a\left(\frac{1}{r^3}
                -y \right)^{1/2} \geq 0;$$ the equality determines photon circular orbits with $E \to \infty$ and $L \to
\pm \infty$. The photon circular orbits of the plus-family are given by the relation $$\label{e51}
  a = a^{(+)}_{\mathrm{ph(1,2)}}(r;y) \equiv \frac{\left(1 -yr^3\right)^{1/2}
                \pm \left(1 -3yr^2 \right)^{1/2}}
                {yr^{3/2}},$$ while for the minus-family orbits they are given by $$\label{e52}
  a = a^{(-)}_{\mathrm{ph(1,2)}}(r;y) \equiv \frac{-\left(1 -yr^3 \right)^{1/2}
                \pm \left(1 -3yr^2 \right)^{1/2}}
                {yr^{3/2}}.$$ A detailed discussion of the photon circular orbits can be found in Refs. .

The behaviour of circular orbits in the field of Kerr black holes ($y=0$) suggests that the plus-family orbits correspond to the co-rotating orbits, while the minus-family circular orbits correspond to the counter-rotating ones. However, this statement is not correct even for Kerr naked-singularity spacetimes with the rotational parameter low enough, where counter-rotating plus-family orbits could exist nearby the ring singularity.[@Stu:1980:BULAI:] In the KdS spacetimes we cannot identify the plus-family circular orbits with purely co-rotating orbits 