While studying for an upcoming exam, I've found the following problem:
Let $X = xz\displaystyle\frac{\partial}{\partial x} + yzxz\displaystyle\frac{\partial}{\partial y}$ and $Y = (1 + z^2)xz\displaystyle\frac{\partial}{\partial z}$ be vector fields in $\mathbb{R}^3$. Determine $\mathcal{U} \subset \mathbb{R}^3$ maximal open set such that $X,Y$ define a foliation in $\mathcal{U}$ and describe such a foliation.
My attempt: I can define a distribution $D: \mathbb{R}^3 \to T_p\mathbb{R}^3$ given by $$D(p) = D_p := \langle X, Y \rangle|_p (= \operatorname{Span}\{X, Y\}|_p).$$ Then, I can show that there is a maximal open set $\mathcal{U} \subset \mathbb{R}^3$ such that $D$ restricted to $\mathcal{U}$ is involutive (namely, $[X,Y] \in \mathfrak{X}(D)$, for $\mathfrak{X}(D)$ vector fields of $D$). If $D$ is involutive then $D$ is integrable, by Frobenius's Theorem, and thus there is some foliation $\mathcal{F}$ such that $D = T\mathcal{F}$.
But I do not know how to proceed from there, i.e., I don't know how to explicitly determine the leaves of $\mathcal{F}$. Also, is this the way to go, or there is a better way to proceed?