I'm attempting to construct a foliation of $S^3$ with leaves diffeomorphic to $S^1$.
My initial thoughts were to construct an involutive distribution generated by $X_1,\dots X_n$ such that $\langle X_1,\dots,X_n\rangle=\operatorname{ker}\omega$, where $\omega=df$ for $f(x,y,z,w)=x^2+y^2$. In this case, I should have $f^{-1}(1)$ is the set $\{(x,y,z,w)\in S^3\mid x^2+y^2=1\}$. Then, it would be great if $$T_{(x,y,z,w)}f^{-1}(1)=\operatorname{ker} df_{(x,y,z,w)}=\operatorname{ker}\omega_p,$$ so it would follow that the integral manifolds of $D=\langle X_1,\dots,X_n\rangle$ are precisely the level sets $x^2+y^2=k$ for $k>0$ in $S^3$.
However, $f^{-1}(1)$ is not (at least by the preimage theorem) an embedded submanifold of $S^3$, because $$df_{(x,y,z,w)}=\begin{pmatrix}2x & 2y & 0 & 0\end{pmatrix}$$ has rank zero at $(0,0,z,w)\in S^3$. Further, while I can show that the distribution spanned by $$X_1=-2y\frac{\partial }{\partial x}+2x\frac{\partial }{\partial y}$$ is in the kernel of $\omega$, this certainly does not constitute the entirely of $\operatorname{ker}\omega$.
Is there a better way to approach problems like these? It feels like I should be using the Frobenius theorem, but I'm having trouble coming up with a good involutive distribution, or a way to produce the associated integral manifolds.