A boundary value problem for the Bitsadze equation
$frac{{partial ^2 }}{{partial bar z^2 }}u(x,y) equiv frac{1}{4}left( {frac{partial }{{partial x}} + ifrac{partial }{{partial y}}} right)^2 u(x,y) = 0$
in the interior of the unit disc is considered. It is proved that the problem is Noetherian and its index is calculated, and solvability conditions for the non-homogeneous problem are proposed. Some solutions of the homogeneous problem are explicitely found.