The Frobenius theorem says that for two vector fields $X,Y$ to be compatible it is sufficient to show that for some functions $\alpha ,\beta$ the following equation holds true \begin{equation} \alpha X + \beta Y + [X,Y] = 0 \end{equation} Assume the two function $\alpha, \beta$ are given, what is the link to the integrability condition ?
As an example, I would like to understand it for the easiest case in $\mathbb{R}^2$. Given the vector fields $X= X^1 \frac{\partial}{\partial X^1}$ and $Y= X^2 \frac{\partial}{\partial X^2}$ with $\alpha=X^2 \frac{\partial X^1}{\partial X^2}$ and $\beta=-X^1 \frac{\partial X^2}{\partial X^1}$ how do I proceed? Do I need to provide more information?
I believe that the solution should be something like the following equation, but unsure as to why. \begin{equation} (X^1 \frac{\partial}{\partial X^1})(\beta X^2)-(X^2 \frac{\partial}{\partial X^2})(\alpha X^1)=0 \end{equation}
In Introduction to smooth manifolds by John M. Lee on page 510 this topic is approached from the angle of Overdetermined Systems of Partial Differential Equations, but I can not bridge the gap between these two topics.
There is also the exterior algebra approach by Robert L. Bryant in Exterior Differential Systems, first mentioned on page 60 and during Cartan-Kähler Theory, where I am hoping to arrive eventually.