Goursat rigid local systems of rank four

We study the general properties of certain rank four rigid local systems considered by Goursat. We analyze when they are irreducible, give an explicit integral description as well as the invariant Hermitian form when it exists. By a computer search we find what we expect are all irreducible such systems all whose solutions are algebraic functions and give several explicit examples defined over the rationals. We also exhibit one example with infinite monodromy as arising from a family of genus two curves.