I have the following exterior differential system for one forms $\alpha, \beta, \gamma$, where the $\theta^i$ are a cotangent basis on $SO(3)$, i.e. they satisfy $d \theta^i = \epsilon_{ijk} \theta^j \wedge \theta^k$ for the antisymmetric tensor $\epsilon$:

$\theta^2 \wedge \alpha = 0$

$\theta^2 \wedge \gamma = 0$

$d \alpha = - \theta^3 \wedge \beta$

$d \beta = \theta^3 \wedge \alpha - \theta^1 \wedge \gamma$

$d \gamma = \theta^1 \wedge \beta$

In addition there is a coordinate $\rho$, i.e. the one forms can be expanded as $\alpha = \alpha_a \theta^a + \alpha_\rho d\rho$ etc.

I guess it doesn't look that imposing (I don't know much of EDSses), but I'm having a seminar talk in a few days, and I've still got plenty to do... It would be awesome if I had a solution to the system above to show the audience! I'd be happy with a proof of existence (a la Frobenius or something I guess) and SOME info on the solution as well or pointers to maybe similar systems in the literature. I'm aware of the text books by Bryant et al.(Google books link) and Ivey&Landsberg (Google books link), but these are thick books and I'm almost outta time!!

I'll think about this myself too, of course, and I'll post here about it too, but now I need to get some sleep...