Problem on finding integral submanifold of a smooth rank-$2$ distribution .

Solution 1:

Here is the trick: by linearity, the distribution spanned by $V_1$ and $V_2$ is the same as the one spanned by $f(x,y,z)V_1(x,y,z)$ and $g(x,y,z)V_2(x,y,z)$ for $f$ and $g$ non-vanishing functions. It follows that the distribution spanned by $V_1$ and $V_2$ is equal to that spanned by $\{\frac{\partial}{\partial x} - \frac{\partial}{\partial y},\frac{\partial}{\partial y}-\frac{\partial}{\partial z}\}$. Since these two latter vector fields commute, it is easy to find the integral submanifold passing through the point $(x_0,y_0,z_0)$: it is parametrized by $(s,t) \mapsto \varphi_t\circ \psi_s (x_0,y_0,z_0)$, where $\varphi$ and $\psi$ are the respective flows.