Let \(\Gamma \in C^ 1\) be a simple closed contour in the complex plane E. Denote by \(G^+\) and \(G^-\) the interior and external domains bounded by the curve \(\Gamma\). Consider the problem: to find a function \(\omega \in C(\bar G^+)\cap C(\bar G^-)\) which is (in a neighbourhood of each point \(z\not\in \Gamma)\) a generalized solution of the equation \[ \partial \omega (z)/\partial \bar z+A(z)\omega (z)+B(z){\bar \omega}(z)=0 \] where \(A(z),B(z)\in L_{p,2}(E)\) \((p>2)\) and their limits \(\omega^+(t)\), \(\omega^-(t)\) satisfy the condition of linear conjugation \[ C(t)\omega^+(t)=d(t)\omega^-(t)+f(t),\quad t\in \Gamma. \] The author considers the case where C(t), d(t) may have a finite number of zeroes of an integral order on \(\Gamma\).