We prove the following theorem (\ref{Tm5}): If $X$ is a nowhere hereditarily
disconnected homogeneous space metrizable by a complete metric,
and $X$ is cleavable over $R$ along every punctured closed connected subset,
then $X$ is locally connected.
Using this result, we establish the next theorem (Theorem \ref{Tm4.17}):
Suppose that $X$ is an infinite homogeneous connected
locally compact metrizable space. Suppose also that $X$ is cleavable over
$R$ along every punctured closed connected subset.
Then $X$ is homeomorphic to the space $R$ of real numbers.