The geometric Cauchy problem forrank-one submanifolds

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