Let $M$ be a compact three-manifold. I am interested in the following equation on $M$:

$ d e^i = \sum_{j,k}^3 \Theta^i_{jk} \, e^j\wedge e^k\, , \qquad i =1,2,3$

together with the following condition:

$0 = [\sum_k^3 L_k e^k]\in H^1(M,\mathbb{R})$

where $(e^1,e^2,e^3)$ is a coframe on $M$ and $\Theta^i_{jk}\in C^{\infty}(M)$ are given functions on $M$ with $L_k := \Theta^{i_0}_{j_0 k}$ for some given $i_0$ and $j_0$ fixed. I want to characterize which Riemannian manifolds $(M,h)$, where:

$h = \sum_k^3 e^k\otimes e^k$

admit solutions to the previous system. This problem should be a particular case of an exterior differential system. However, I fail to see how to apply the theory correctly. First of all, I guess that we have to consider a larger manifold such that solutions to the system are integral submanifolds. I guess this should be the frame bundle of $M$. In any case, I don't fully understand how to apply the general theory of exterior differential systems, and which type of results this theory can provide in this problem. Also, it is not clear to me which results of the theory are purely local. Any help will be very welcome!

Thanks a lot!