Integral equations for the H- X- and Y-functions

We come back to a non linear integral equation satisfied by the function H, which is distinct from the classical H-equation. Established for the first time by Busbridge (1955), it appeared occasionally in the literature since then. First of all, this equation is generalized over the whole complex plane using the method of residues. Then its counterpart in a finite slab is derived; it consists in two series of integral equations for the X- and Y-functions. These integral equations are finally applied to the solution of the albedo problem in a slab.