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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.

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There exists a theory of integrability for differential equations on vector bundles over infinite dimensional jet bundles or, more generally, diffities. Although several vector fields on such structures are not integrable, relevant vector fields, like by the operators $D_\alpha$ and other evolutionary vector fields, can be integrated. This makes simpler to investigate the integrability of such a type of differential equations. More general results are based upon the cohomological theory of infinite-dimensional jet bundles, diffieties and their C-spectral sequences. Some references can be found below and in the literature therein.

J. Krasilshchik and A. Verbovetsky, Geometry of jet spaces and integrable systems, Journal of Geometry and Physics Volume 61, Issue 9, September 2011, Pages 1633-1674.

A. M. Vinogradov, A spectral sequence associated with a nonlinear differential equation and algebro-geometric foundations of Lagrangian field theory with constraints, Soviet Math. Dokl. 19, 144–148 (1978).

D. Gessler, On the Vinogradov C-spectral sequences for determined systems of differential equations, Differential Geometry and its applications 7, 303-324 (1997).

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Section 4 of the paper "Geometry of Differential Equations" by B. Kruglikov and V. Lychagin (IHES/M/07/04) states that formal integrability & analyticity are sufficient for the existence of a solution.

Equations of the specific type shown in your question have been discussed extensively by S. Igonin (e.g. https://arxiv.org/abs/1703.07217) and J. Krasil'shchik

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