The generalized Higgs fiber bundles

Question

Let $(M,J)$ be a complex manifold. A generalized Higgs fiber bundle is an holomorphic fiber bundle $E$ over $M$ with holomorphic $\phi_i \in \Lambda^1 (M) \otimes End (E)$, $1\leq i\leq k$, such that:

$$ \phi_i \wedge \phi_j=0$$

$\forall i,j$. I define:

$$\bar D_i=\bar \partial +\phi_i$$

Then, the condition is $\forall i,j$:

$$\bar D_i \circ \bar D_j=- \bar D_j \circ \bar D_i$$

Can we define the moduli spaces of the generalized Higgs fiber bundles?