# Solving differential forms equations

I have this couple of equations in differential forms language:

$$\Delta z^i \star {\bf{1}}+ G^{i\bar{p}} (\partial_j G_{k\bar{p}} ) dz^j \wedge \star dz^k + G^{i\bar{j}} (\partial_{\bar{j}} G_{k\bar{l}} ) dz^k \wedge \star dz^{\bar{l}} =0$$

and

$$\Delta z^{\bar{i}} \star {\bf{1}}+ G^{p\bar{i}} (\partial_{\bar{j}} G_{\bar{k} p} ) dz^{\bar{j}} \wedge \star dz^{\bar{k}} + G^{\bar{i}j} (\partial_j G_{k\bar{l}} ) dz^k \wedge \star dz^{\bar{l}} =0$$

They are the equation of z and its complex conjugate on complex coordinates.

Can I solve them in Mathematica to get the form $$z^i$$ and the matrix $$G_{ij}$$ , even numerically?