The equation is
$$S(z,t)=\int_a^b \int_c^d f(z,u) g(t,s) S(u,s) \, du \, ds$$
where $S$ is the unknown function and $f,\ g$ are fixed from the outset.
I can approximate solutions in some special cases with numerical methods but I'm interested in knowing if there's literature on this equation I could look to for more information.