When are the coordinate functions of a distribution independent of the leaf in the foliation?

Question

By the Frobenius theorem, given an involutive distribution $\{X_i\}_{i = 1}^m$ on a manifold $M^n$ we can find a coordinate patch $(x^i)$ s.t. for some $q$ the set $x^{q+1} = \cdots = x^n = \text{constant}$ corresponds to a submanifold $U \subset M$ with $TU = \text{span}\{X_i\}_{i=1}^m$. This means that $$ X_i = \sum_{j = 1}^q a_i^j(x^1, \dots, x^n) \partial_j. $$ Under which conditions do we know that $a_i^j(x^1, \dots, x^n) = a_i^j(x^1, \dots, x^q)$?