Regular rotating de Sitter-Kerr black holes and solitons

We study the basic generic properties of the class of regular rotating solutions asymptotically Kerr for a distant observer, obtained with using the Gurses-Gursey algorithm from regular spherically symmetric solutions specified by T-t(t) = T-r(r) which belong to the Kerr-Schild metrics. All regular solutions obtained with the Newman-Janis complex translation from the known spherical solutions, belong to this class. Spherical solutions with T-t(t) = T-r(r) satisfying the weak energy condition (WEC), have obligatory de Sitter center. Rotation transforms the de Sitter center into the interior de Sitter vacuum disk. Regular de Sitter-Kerr solutions have at most two horizons and two ergospheres, and two different kinds of interiors. In the case when an original spherical solution satisfies the dominant energy condition, there can exist the interior de Sitter vacuum S-surface which contains the de Sitter disk as a bridge. The WEC is violated in the internal cavities between the S-surface and the disk, which are filled thus with a phantom fluid. In the case when a related spherical solution violates the dominant energy condition, vacuum interior of a rotating solution reduces to the de Sitter disk only.