Construction of “Hauptfunktion”, solution of the equations of Schwarz and Puchs for a surface of zero genus by the methods of spectral theory of automorphic functions

Fuchsian and Schwarzian equations with $n$ singularities on a surface of zero genus are studied. The methods of spectral theory of automorphic functions are used to construct explicity the Klein invariant for an arbitrary Fuchsian group of genus 0 with fundamental domain of finite area. This result is used to derive explicit formula expressing singularities and accessory parameters for Fuchsian and Schwarzian equations in terms of the corresponding Fuchsian group.