  We demonstrate that five-dimensional, asymptotically flat, stationary and
biaxisymmetric, vacuum black holes with lens space $L(n, 1)$ topology,
possessing the simplest rod structure, do not exist. In particular, we show
that the general solution on the axes and horizon, which we recently
constructed by exploiting the integrability of this system, must suffer from a
conical singularity on the inner axis component. We give a proof of this for
two distinct singly spinning configurations and numerical evidence for the
generic doubly spinning solution.
