  Quintessential dark energy with density $\rho$ and pressure $p$ is governed
by an equation of state of the form $p=-\omega_{q}\rho$ with the quintessential
parameter $\omega_q\in(-1;-1/3)$. We derive the geometry of quintessential
rotating black holes, generalizing thus the Kerr spacetimes. Then we study the
quintessential rotating black hole spacetimes with the special value of
$\omega_q = -2/3$ when the resulting formulae are simple and easily tractable.
We show that such special spacetimes can exist for dimensionless quintessential
parameter $c<1/6$ and determine the critical rotational parameter $a_0$
separating the black hole and naked singularity spacetime in dependence on the
quintessential parameter $c$. For the spacetimes with $\omega_q = 2/3$ we
present the integrated geodesic equations in separated form and study in
details the circular geodetical orbits. We give radii and parameters of the
photon circular orbits, marginally bound and marginally stable orbits. We
stress that the outer boundary on the existence of circular geodesics, given by
the so called static radius where the gravitational attraction of the black
hole is balanced by the cosmic repulsion, does not depend on the dimensionless
spin of the rotating black hole, similarly to the case of the Kerr-de Sitter
spacetimes with vacuum dark energy. We also give restrictions on the
dimensionless parameters $c$ and $a$ of the spacetimes allowing for existence
of stable circular geodesics.
