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

We study invertible matrix solutions $A$ to the equation $A^{-1}\overline\partial A=\omega^{(0,1)}$ on a small open subset $U$ of the closure $\overline M$ of a domain $M\subset{\mathbf C}^n$, where $\omega^{(0,1)}$ is a matrix of $(0,1)$ forms on $\overline M$ satisfying the formal integrable condition $\overline{\partial}\omega^{(0,1)}=\omega^{(0,1)}\wedge\omega^{(0,1)}$. 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.