Let $\mathbb{M}$ be a $n+m$ dimensional manifold. Consider on $\mathbb{M}$ a rank $n$ sub-bundle $\mathcal{H}$ of the tangent bundle. We assume that $\mathcal{H}$ is endowed with a fiber wise inner product $g_\mathcal{H}$.

Under which conditions on $(\mathbb{M}, \mathcal{H}, g_\mathcal{H})$ can we find a totally geodesic Riemannian foliation $(\mathcal{F},g)$ on $\mathbb{M}$ such that:

1) $g$ is bundle-like,

2) $g$ restricts to $g_\mathcal{H}$ on $\mathcal{H}$

3) The leaves of the foliation are orthogonal to $\mathcal{H}$ ?