Given a point $p_0$ in $\mathbb{R}^3$ and two tanget vectors fields $v_1$ and $v_2$, which satisfy $$[v_1,v_2]\neq a(p)v_1+b(p)v_2$$ for arbitrary function $a(p),b(p)$, namely, $\{v_1,v_2\}$ is not integrable.
Take the exponential map $\Phi(t_1,t_2)=\exp(t_1v_1+t_2v_2)$ with the initial point $p_0$, where $t_1,t_2$ are constants.
Question:
Is $\Phi(t_1,t_2)$ a function? If $\{v_1,v_2\}$ are integrable, but non-commutative, $\Phi(t_1,t_2)$ is a function ($\partial_{t_1}\partial_{t_2}\Phi\neq \partial_{t_2}\partial_{t_1}\Phi$)?
The set of $\exp(t_1v_1+t_2v_2)$ for all $t_1,t_2\in \mathbb{R}$ is a surface or not? What is the difference of the set in the cases of $\{v_1,v_2\}$ integrable or non-integrable?