  We study the decay $B^{+} \to K^+ K^- \pi^+$ and investigate the angular
distribution of $K^{+}K^{-}$ pairs with invariant mass below $1.1$ GeV/$c^2$.
This region exhibits both a strong enhancement in signal and very large direct
$CP$ violation. We construct a coherent sum model for the angular distribution
of $S$- and $P$-wave, and report the ratio of their amplitudes, the relative
phase and the forward-backward asymmetry. We also report absolute differential
branching fractions and direct $CP$ asymmetry for the decay in bins of
$M_{K^+K^-}$ and the differential branching fractions in bins of
$M_{K^+\pi^-}$. The results are based on a data sample that contains
$772\times10^6$ $B \bar{B}$ pairs collected at the $\Upsilon(4S)$ resonance
with the Belle detector at the KEKB asymmetric-energy $e^+ e^-$ collider. The
measured overall branching fraction and the direct $CP$ asymmetry are
$(5.38\pm0.40\pm0.35)\times 10^{-6}$ and $-0.170\pm0.073\pm0.017$,
respectively, where the first uncertainties are statistical and the second are
systematic.
