  The high-spin rotational bands in odd-$Z$ nuclei $^{159}$Ho ($Z=67$) are
investigated using the cranked shell model with the pairing correlations
treated by a particle-number conserving method, in which the blocking effects
are taken into account exactly. The experimental moments of inertia and
alignments and their variations with the rotational frequency $\hbar\omega$ are
reproduced very well by the calculations. The splitting between the signature
partners of the yrast band $7/2^-[523]$ is discussed and the splitting of the
excited band $7/2^+[404]$ above $\hbar\omega \sim 0.30$~MeV is predicted due to
the level crossing with $1/2^+[411]$. The calculated $B(E2)$ transition
probabilities are also suggested for future experiments.
