The Lee equations

Question

Let $(M,g)$ be a riemannian four-manifold and $\omega$ an exterior form, $\theta$ a 1-form. I define the Lee equations:

$$d\omega = \theta \wedge \omega$$

$$d^* \omega= \iota (\theta^*)(\omega)$$

$$d\theta_+=0$$

with $\iota (X)(\omega) (Y_i)=\omega (X,Y_i)$.

The gauge group is the inversible functions, it acts $f.(\omega,\theta)=(f \omega,\theta +df/f)$.

Have we a good moduli space?