Poincare-Steklov Integral equation

Integral equation under consideration connects via the spectral parameter the integral operator with Cauchy kernel and the integral operator with Grunsky kernel. Functional parameter of the equation that defines Grunsky kernel is a variable change on the interval of integration. Equation arises in the dimensional reduction of the following boundary value problem:
A flat domain is divided into two pieces by the interface. We are looking for a continuous function, harmonic in each subdomain and satisfying Dirichlet condition on the outer boundary. At the interface the values ​​of the normal derivatives differ by a factor, the spectral parameter. It will be shown how to explicitly solve the spectral problem for the integral equation in the simplest case when the functional parameter is a degree two rational function.
[1] AB Bogatyrev Geometric method for solving Poincare-Steklov integral equation // Math.Notes 63: 3 (1998) [2] AB Bogatyrev Integral equations PS and the Riemann monodromy problem // Func.An.&Appl. 34: 2 (2000) [3] AB Bogatyrev Integral equations PS-3 and projective structures on Riemann surfaces // Sbornik: Math. 192: 4 (2001) [4] Bogatyrev A.B. Pictorial Representations of antisymmetric Eigenfunctions of PS-3 integral Equations // Math. Physics, Analysis and Geometry (Springer), 13 (2010), 105-143.