Regularization of a three-element functional equation

In this paper we study the three-element functional equation
$$ (V\Phi)(z)\equiv\Phi(iz)+\Phi(-iz)+G(z)\Phi(\frac1z)=g(z),\qquad z\in R, $$
subject to
$$ R\colon |z|<1,\quad|\arg z|<\frac\pi4. $$

We assume that the coefficients $G(z)$ and $g(z)$ are holomorphic in $R$ and their boundary values $G^+(t)$ and $g^+(t)$ belong to $H(\Gamma)$, $G(t)G(t^{-1})=1$. We seek for solutions $\Phi(z)$ in the class of functions holomorphic outside of $\overline R$ such that they vanish at infinity and their boundary values $\Phi^-(t)$ also belong to $H(\Gamma)$.
Using the method of equivalent regularization, we reduce the problem to the 2nd kind integral Fredholm equation.