  No Kerr-like exact solution has yet been found in Chern-Simons modified
gravity. Intrigued by this absence, we study stationary and axisymmetric
metrics that could represent the exterior field of spinning black holes. For
the standard choice of the background scalar, the modified field equations
decouple into the Einstein equations and additional constraints. These
constraints eliminate essentially all solutions except for Schwarzschild. For
non-canonical choices of the background scalar, we find several exact solutions
of the modified field equations, including mathematical black holes and
pp-waves. We show that the ultrarelativistically boosted Kerr metric can
satisfy the modified field equations, and we argue that physical spinning black
holes may exist in Chern-Simons modified gravity only if the metric breaks
stationarity, axisymmetry or energy-momentum conservation.
