The geometric Cauchy problem for rank-one submanifolds

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.