Change of coordinates in Frobenius Theorem.

Question

I was reading a proof of Frobenius Theorem but I can't understand the last part on changing coordinates.

(The following proof of the Theorem assumes involute property and wants to prove integrability.)

In the last paragraph, how is $u^i$ constructed? By lemma 2, we can pick coordinates $\bar u^i$ so that $Y=\frac{\partial}{\partial \bar u^r}$, but I'm not sure how then we can construct $u^i$.