The geometric Cauchy problem for constant-rank submanifolds

Given a smooth $s$-dimensional submanifold $S$ of ${\mathbb{R}}^{m+c}$ and a smooth distribution $D\supset TS$ of rank $m$ along $S$, we study the following geometric Cauchy problem: to find an $m$-dimensional rank-$s$ submanifold $M$ of ${\mathbb{R}}^{m+c}$ (that is, an $m$-submanifold with constant index of relative nullity $m-s$) such that $M\supset S$ and $TM{\mathrm{\mid }}_S=D$. In particular, under some reasonable assumption and using a constructive approach, we show that a solution exists and is unique in a neighborhood of $S$.