$\mathbf {The \ Problem \ is}:$ Let $V_{(x,y,z)}$ be a rank-$2$distribution in $\mathbb R^3$ spanned by $V_1=(x^2+1)(\frac{\partial}{\partial{x}}-\frac{\partial}{\partial{y}})$ & $V_2=\frac{\frac{\partial}{\partial{y}}-\frac{\partial}{\partial{z}}}{x^2+y^2+z^2+1}.$
Show it is a $2$-dimensional smooth distribution & find the integral submanifold of $V$ passing via the origin .
$\mathbf {My \ approach}:$ The smooth part is okay .
For 2nd part, I was trying to find out integral curves of $V_1$ & $V_2$ via $(0,0,0).$
For $V_1$,it's $t \mapsto (\tan t,-\tan t,0)$ but
if $\gamma(t)=(\gamma_1(t),\gamma_2(t),\gamma_3(t))$ be I.C. of $V_2$ via origin then :
$\frac{d\gamma_2}{dt} =\frac{1}{1+(\gamma_2)^2+(\gamma_3)^2}$ & $\frac{d\gamma_3}{dt} =-\frac{1}{1+(\gamma_2)^2+(\gamma_3)^2}$ .
Is this approach right ?
If yes, then after this I need a moderate hint & I'm a beginner in differential forms . Thanks in advance .
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.