  We study various mathematical aspects of the charged rotating black hole with
two equal-magnitude angular momenta in five dimensions. We introduce a
coordinate system that is regular on the horizon and in which Einstein-Maxwell
equations reduce to an autonomous system of ODEs. Employing Bondi and
Kruskal-like coordinates, we analyze the geometric regularity of the black hole
metric at infinity and the horizon, respectively, and the well-posedness of the
corresponding boundary value problem.
  We also study the algebraic types of the electromagnetic and curvature
tensors. While outside the horizon the electromagnetic and Ricci tensors are of
type D, the Weyl tensor is algebraically general. The Weyl tensor simplifies to
type~II on the horizon and type~D on the bifurcation sphere. These results
imply inconsistency of the metric with the Kerr--Schild form with a geodesic
Kerr-Schild vector. This feature is shared by the four-dimensional Kerr-Newman
metric and the vacuum Myers-Perry or charged Schwarzschild-Tangherlini
geometries in arbitrary dimension, but hence not by the black hole we have
considered here.
