  Given a smooth distribution $\mathscr{D}$ of $m$-dimensional planes along a
smooth regular curve $\gamma$ in $\mathbb{R}^{m+n}$, we consider the following
problem: to find an $m$-dimensional rank-one submanifold of $\mathbb{R}^{m+n}$,
that is, an $(m-1)$-ruled submanifold with constant tangent space along the
rulings, such that its tangent bundle along $\gamma$ coincides with
$\mathscr{D}$. In particular, we give sufficient conditions for the local
well-posedness of the problem, together with a parametric description of the
solution.
