  In this paper, a coherent study of the $e^+e^-$ annihilation into
$K^+K^-\pi^0$, $K^0_SK^0_L\pi^0$ and $K^0_SK^\pm\pi^\mp$ is carried out within
the framework of resonance chiral theory. The amplitudes are fixed by fitting
to the experimental cross-section and invariant mass spectrum. With these
amplitudes, one can calculate the hadronic vacuum polarization form factors of
these processes. The leading order contributions of $\sigma(e^+e^- \to
K\bar{K}\pi)$ to the anomalous magnetic moment of the muon, $(g-2)_\mu$, is
obtained as $a_\mu^{\rm HVP,LO}(e^+e^- \to
K\bar{K}\pi)=(3.07\pm0.07)\times10^{-10}$ up to $E_{\rm cm}=2.3$ GeV.
