Let $M$ be a compact $n$-dimensional $C^\infty$-manifold, and $(U;x_1,...,x_n)$ be a local coordinate system on it.
Consider $C^\infty$-vector fields on $M$ defined by:
$X_i(x)=\sum_{j=1}^n a_{ij}(x)\frac{\partial}{\partial x_j}. \quad (x\in U, \ i=1,...,n)$ ... (1)
Assume that each vector field is linearly independent, meaning that for every $p\in U$, $X_i(p)$ ($i=1,...,n$) is linearly independent in $T_p(M)$.
For simplicity, let's assume that each $a_{ij}(x)$ is nonzero.
In this case, there exists a local coordinate system $(U;y_1,...,y_n)$ such that:
$X_i(x)=\frac{\partial}{\partial y_i}. \quad (x\in U, \ i=1,...,n)$ ... (2)
(I am not aware of a rigorous proof for this, but I believe it should be true. If there are references with a proof, I would appreciate it if you could let me know.)
(Addendum: Generally, (2) does not hold. I want to know the conditions under which it holds. Please consider the following statements in a situation where (2) holds.)
Then, there exists a coordinate transformation from $(x_1,...,x_n)$ to $(y_1,...,y_n)$ such that:
$\frac{\partial}{\partial y_i}=\sum_{j=1}^n \frac{\partial x_j}{\partial y_i}\frac{\partial}{\partial x_j}. \quad (i=1,...,n)$ ... (3)
By combining (1), (2), and (3), we obtain:
$a_{ij}(x_1,...,x_n)=\frac{\partial x_j}{\partial y_i}. \quad (i,j=1,...,n)$
How can one determine $y_i=y_i(x_1, ..., x_n)$ ($i=1,...,n$) from such a system of partial differential equations?
Moreover, can it be asserted that a unique solution exists in the first place?