Is there any result on the existence of solutions of differential equations of the form $$ D_\alpha\Phi([u])=U_\alpha([u])\Phi([u]), $$ where $[u]$ is an element of an infinite dimensional bundle $J^\infty$, the $D_\alpha$ is the total derivative relative to the base variable of the jet bundle, $\Phi([u])$ takes values in a matrix Lie group $G$, and $U_\alpha([u])$ takes values in its Lie algebra $\mathfrak{g}$?
Of course, the above differential equations must satisfy an integrability condition to have a solution, namely $D_\alpha D_\beta\Phi([u])=D_\beta D_\alpha\Phi([u])$. Nevertheless, is this condition sufficient? It is obvious that the Frobenius theorem does not apply here as $J^\infty$ is infinite-dimensional.
