The geometric Cauchy problem for rank-one submanifolds

Given a smooth distribution $\mathcal{D}$ 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 $\mathcal{D}$. In particular, we give sufficient conditions for the local well-posedness of the problem, together with a parametric description of the solution.