A function theory method in boundary value problems in the plane. I. The smooth case

A general (not necessarily local) boundary value problem is considered for an elliptic $(l\times l)$ system on the plane of $n$th order containing only leading terms with constant coefficients. By a method of function theory developed for elliptic $(s\times s)$ systems of first order
$$ \frac{\partial\Phi}{\partial y}-J\frac{\partial\Phi}{\partial x}=0 $$
with a constant triangular matrix $J=(J_{ij})_1^s$, $\operatorname{Im}J_{ij}>0$; this problem is reduced to an equivalent system of integrofunctional equations on the boundary. In particular, a criterion that the problem be Noetherian and a formula for its index are obtained in this way. All considerations are carried out in the smooth case when the boundary of the domain has no corner points, while the boundary operators act in spaces of continuous functions.