  Let $V^n$ be an open manifold of non-negative sectional curvature with a soul
$\Sigma$ of co-dimension two. The universal cover $\tilde N$ of the unit normal
bundle $N$ of the soul in such a manifold is isometric to the direct product
$M^{n-2}\times R$. In the study of the metric structure of $V^n$ an important
role plays the vector field $X$ which belongs to the projection of the vertical
planes distribution of the Riemannian submersion $\pi:V\to\Sigma$ on the factor
$M$ in this metric splitting $\tilde N=M\times R$. The case $n=4$ was
considered in [GT] where the authors prove that $X$ is a Killing vector field
while the manifold $V^4$ is isometric to the quotient of $M^2\times
(R^2,g_F)\times R$ by the flow along the corresponding Killing field. Following
an approach of [GT] we consider the next case $n=5$ and obtain the same result
under the assumption that the set of zeros of $X$ is not empty. Under this
assumption we prove that both $M^3$ and $\Sigma^3$ admit an open-book
decomposition with a bending which is a closed geodesic and pages which are
totally geodesic two-spheres, the vector field $X$ is Killing, while the whole
manifold $V^5$ is isometric to the quotient of $M^3\times (R^2,g_F)\times R$ by
the flow along corresponding Killing field.
