Integrability of Koszul connections on complex vector bundles over domains in ${\mathbf{C}}^n$

We study invertible matrix solutions $A$ to the equation $A^{-1}\bar{\mathrm{\partial }}A={\omega }^{(0,1)}$ on a small open subset $U$ of the closure $\bar{M}$ of a domain $M\subset {\mathbf{C}}^n$, where ${\omega }^{(0,1)}$ is a matrix of $(0,1)$ forms on $\bar{M}$ satisfying the formal integrable condition $\bar{\mathrm{\partial }}{\omega }^{(0,1)}={\omega }^{(0,1)}\wedge {\omega }^{(0,1)}$. For a $C^2$ domain $M$ that is either strongly pseudoconvex or has at least $3$ negative Levi eigenvalues at a boundary point contained in $U$, we obtain existence and sharp regularity of the solutions.