No reference whatsoever to the Picard-Lindelof or Frobenius theorem is required.  Please see the definition of tangent space.  It doesn't matter whether you get a particular flow through the point in question, only that there exists one to first order.  And then it's just, pick a flow, any flow.  Those who know about such things (myself included) are aware of these theorems, but they add needless clutter to the article.  Simply things as far as possible, and no further.     