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.