Say we are working in the following toy situation. Let $X$ and $Y$ be vector fields over a smooth manifold $M$, and say we have some equation involving the Lie bracket, e.g. $[X,Y] = X+Y$. Say $Y$ is fixed, and we wish to solve for all $X$ that satisfies this equation.
This becomes a PDE in the following sense, as far as I understand. First we work locally and write $$ X = \sum_{i=1}^n X^i \partial_i, ~~~~ Y = \sum_{i=1}^n Y^i \partial_i, ~~~~ [X,Y] = \sum_{i=1}^n \sum_{j=1}^n (X^j \partial_j Y^i - Y^j \partial_j X^i)\partial_i $$ over some chart $U$, where $X^i, Y^i$ are functions and $\partial_i = \frac{\partial}{\partial x^i}$ is the associated local basis of the tangent bundle.
Then, plugging the above in the equation, we isolate the functions in front of each $\partial_i$ that appears and we get for fixed $i$, $$ \sum_{j=1}^n (X^j \partial_j Y^i - Y^j \partial_j X^i) = X^i + Y^i. $$ So in particular, this is a system of PDEs where we wish to solve for the functions $X^j$ over a domain that is diffeomorphic to $\mathbb{R}^n$. Thus locally we "forget" about the manifold structure and just work in the flat case (if I understand this part correctly).
But because we are forgetting about the global structure, working locally like this seems very insufficient to finding a solution $X$ over all of $M$. How do we go from local to global to solve for the vector field $X$?
EDIT: I'm more interested in the question of the existence of a solution $X$, rather than an explicit solution.
