On regular frames near rotating black holes

We consider the metric of a generic axially symmetric rotating stationary black hole. The general approach is developed that enables us to construct coordinate frame regular near the horizon. As explicit examples, the Kerr and Kerr-Newmann-(anti-)de Sitter metrics are considered. It is shown how the rotational versions of the Painleve'-Gullstrand and Doran coordinates appear in this scheme as particular cases. For the 2+1 version of the metric the direct generalization of the Lema\^itre coordinate system is obtained. It is shown that the possibility of introducing a regular frame is indirectly related to the constancy of a black hole angular velocity and the rate with which the metric coefficient responsible for the rotation of spacetime, tends to it.