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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?

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