  Let $\pi:E\to M$ be a vector bundle over a simply connected manifold and
$\nabla$ a linear connection in $\pi$. Let $\sigma: U \rightarrow E$ be a
$\nabla$-parallel section of $\pi$ defined on a connected open subset $U$ of
$M$. We give sufficient conditions on $U$ in order to extend $\sigma$ to the
whole $M$. We mainly concentrate to the case when $M$ is a $2$-dimensional
simply connected manifold.
