I'm reading an article that says we can garantee a solution u=u(x,t) with $(x,t)$ in a neighboor of origin of $(0,0) \in \mathbb R^{m+n}$ for the problem $$ L_j u = 0, j=1, \ldots n, $$ where $$L_j = \frac{\partial}{\partial t_j} + \sum_{k=1}^m A_k(t,x) \frac{\partial}{\partial x_k}$$ are 2-2 commutative analytic vector fields with the condition $$u(x,0) = F(x),$$ $F$ also analytic in neighboor of origin of $\mathbb R^m$. I'm trying to use the Cauchy Kovalevskaia Theorem but the things are not filling.
I was trying to solve, for each $j=1,\ldots n$, the problem $$ L_j u = 0, j=1, \ldots n, \ \ \ u(x,t_1,\ldots,0,\ldots,t_n) = F(x), $$ or different inicial data, and try to create our desire solution with sums, product or convolution of that solutions, but I failed.
