Based on the Newman-Janis algorithm the Ayón-Beato-García spacetime metric [1] of the regular spherically symmetric, static and charged black hole has been converted into rotational form. It is shown that the derived solution for rotating regular black hole is regular and the critical value of the electric charge Q for which two horizons merge into one sufficiently decreases in the presence of nonvanishing angular momentum a of the black hole.

Quintessential dark energy with density ρ and pressure p is governed by an equation of state of the form p = −ωqρ with the quintessential parameter ωq ∈ (−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 ωq = −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 a0 separating the black hole and naked singularity spacetime in dependence on the quintessential parameter c. For the spacetimes with ωq = 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.