  In 1963 Ezra Ted Newman and his two students Louis A. Tamburino, and Theodore
W. J. Unti, proposed a deformation of the Shwarzschild spacetime that made it
twisting. In the cosmological context, an equivalent solution had been found
earlier, in 1951, by Abraham Haskel Taub. The problem that these solutions have
is a conical singularity along the symmetry axis at all distances from the
origin. In 1969 Misner proposed a non-singular interpretation of Taub-NUT
spacetimes. We extend and refine his method to include a broader family of
solutions and completely solve the outstanding issue of a non-singular
extension of the Kerr-NUT- (anti) de Sitter solutions to Einstein's equations.
Our approach relies on an observation that in 2 dimensional algebra of Killing
vector fields there exist 2 distinguished vector fields that may be used to
define $U(1)$-principal bundle structure over the non-singular spaces of
non-null orbits. For all admissible parameters we derive appropriate Killing
vector fields and discuss limits to spacetimes with less parameters. The global
structure of spacetime, together with non-singular conformal geometry of the
infinities is presented and (possibly also projectively non-singular) Killing
horizons is presented.
