If one has two vector fields $X$ and $Y$ in $\mathbb{R}^4$ whose coefficients are indeterminate functions (some coefficients are being forced to be linearly related to others), then the condition that the bracket $[X,Y]$ vanishes gives a system of nonlinear first order PDEs. Non linear PDEs are notoriously hard to work with as I understand, but I am suspicious that this bracket condition is common enough that there might have been real progress made in existence type theorems for solving them (in my situation, my system is under-determined, but being nonlinear, I don't have a good way to prove existence of any solutions at all - and I would like to prove existence of solutions under a variety of initial conditions).
Can anybody give me some idea of what sort of existence theorems for solving this general type of PDE might exist, and where I might find references to them? This is new territory for me and I don't have access to MathSciNet.