Generalized analytic functions of several variables

In this paper the system of differential equations
$$ \frac{\partial u}{\partial\bar z_k}-\overline{a_ku}=f_k,\qquad k=1,2,\dots,n $$
is studied in a domain in $\mathbf C^n$. Differential relations between the coefficients are indicated under which the homogeneous system ($f_k=0$) has infinitely many linearly independent local solutions at each point. Global solvability of the system is studied. It is proved that this problem, and also the description of global solutions of the homogeneous system, reduces to analogous questions for a connection of the type $\partial u/\partial\bar z=\overline{au}+bu$ in a line bundle over a Riemann surface, related to the original system in a natural way. Using this reduction, in some cases one can establish that the index of the operator corresponding to this system is finite and find a formula for it.
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