Poincare-Steklov integral equation: Episode 1

This integral equation relates (via the spectral parameter) the integral operator with Cauchy kernel and the integral operator with Grunskii kernel. The functional parameter of the equation that defines Grunskii kernel is the variable change on the finite interval of integration. The equation arises in the reduction of some boundary problem for harmonic functions with a spectral parameter in boundary conditions. It will be shown how to explicitly solve the spectral problem for the integral equation in the simple case when the functional parameter is a quadratic polynomial.