Solution in closed form of an integral equation of convolution type in a hyperelliptic case (Q1589164)

The author considers a Wiener-Hopf integral equation on the real line perturbed by terms with conjugation: \((\lambda_1+K_1)\theta_+\varphi+(\lambda_3+K_3)\theta_+\overline{\varphi} +(\lambda_2+K_2)\theta_-\varphi+(\lambda_4+K_4)\theta_-\overline{\varphi}=f(x)\), \(-\infty <x <\infty,\) where \((K_j\varphi)(x)=(K_j\ast\varphi)(x)\) and \(\theta_\pm (x)=\frac 12(1\pm \text{sign } x)\). Under some assumptions some cases are found when this equation may be solved in closed form.