On applications of summary equation induced by quadrilateral

Let $D$ be an arbitrary quadrilateral. On this quadrilateral, we consider a linear summary four-elements equation with the class of solutions holomorphic outside $D$ and vanishing at infinity. Their boundary the values satisfy the Hölder condition on each compact set containing no peaks. If the peaks are present, at them, at most logarithmic singularities are admitted. The free term is holomorphic on $D$ and its boundary value satisfies the Hölder condition. It is not assume to admit an analytic continuation through some segment of the boundary, that is, the solution and the free term belong to different classes of holomorphic functions. In order to regularize this equation on the boundary of the quadrilateral, we introduce a piece-wise linear Carleman translation mapping each side into itself by changing the orientation. This translation is discontinuous at the vertices and has fixed points at the centers of the side. The solution can be represented as a Cauchy type integral over a boundary with an unknown density invariant with respect to the shift on one pair of adjacent sides and anti-invariant on the other pair. We show that the regularization is equivalent. In some particular cases the obtained Fredholm equation is solvable. As an example, we choose an quadrilateral with a straight angle. We construct a system of entire functions of a completely regular growth biorthogonal to the system of powers with a piece-wise quasi-polynomial weight.